Shell method with two functions of x

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.210
This right here is a
solid of revolution

00:00:04.210 --> 00:00:07.590
whose volume we were able to
figure out in previous videos,

00:00:07.590 --> 00:00:11.070
actually in a different
tutorial, using the disk method

00:00:11.070 --> 00:00:13.352
and integrating in terms of y.

00:00:13.352 --> 00:00:14.810
What we're going
to do right now is

00:00:14.810 --> 00:00:17.537
we're going to find the same
volume for the same solid

00:00:17.537 --> 00:00:20.120
of revolution, but we're going
to do it using the shell method

00:00:20.120 --> 00:00:22.250
and integrating
with respect to x.

00:00:22.250 --> 00:00:24.920
So what we do is we have
the region between these two

00:00:24.920 --> 00:00:26.620
curves, y is equal
to square root of x

00:00:26.620 --> 00:00:28.130
and y is equal to x squared.

00:00:28.130 --> 00:00:31.880
And we're going to rotate it
around the vertical line x

00:00:31.880 --> 00:00:32.740
equals 2.

00:00:32.740 --> 00:00:34.547
And we're going to do
that at the interval

00:00:34.547 --> 00:00:37.130
that we're going to rotate this
space between these two curves

00:00:37.130 --> 00:00:42.150
is the interval when square root
of x is greater than x squared.

00:00:42.150 --> 00:00:45.920
And so it's between 0 and 1.

00:00:45.920 --> 00:00:48.690
And so let's try to do
it with the shell method.

00:00:48.690 --> 00:00:54.090
And so to do that, what we do
is we want to construct a shell.

00:00:54.090 --> 00:00:55.880
Let me do this in
a different color.

00:00:55.880 --> 00:00:58.100
Let's imagine a rectangle
right over here.

00:01:01.630 --> 00:01:05.250
It has width dx.

00:01:05.250 --> 00:01:07.937
and its height is the difference
of these two functions.

00:01:07.937 --> 00:01:09.770
And so if I were to
draw it right over here,

00:01:09.770 --> 00:01:11.186
it would look
something like this.

00:01:14.480 --> 00:01:18.310
It'd be there, and
then it is a shell,

00:01:18.310 --> 00:01:20.240
it's kind of a
hollowed-out cylinder.

00:01:20.240 --> 00:01:23.340
So it would look
something like this.

00:01:26.620 --> 00:01:28.340
Just like that.

00:01:28.340 --> 00:01:32.100
And it has some depth,
that's what the dx gives us.

00:01:32.100 --> 00:01:36.182
So we have the depth that
looks something like that.

00:01:36.182 --> 00:01:37.890
And then let me shade
it in a little bit,

00:01:37.890 --> 00:01:42.990
just so we can see a
little bit of its depth.

00:01:42.990 --> 00:01:44.580
So when you rotate
this rectangle

00:01:44.580 --> 00:01:47.790
around the line x equals 2,
you get a shell like this.

00:01:47.790 --> 00:01:49.540
So let's think about
how we can figure out

00:01:49.540 --> 00:01:51.317
the volume of this shell.

00:01:51.317 --> 00:01:53.150
Well, we've already
done this several times.

00:01:53.150 --> 00:01:54.983
The first thing we might
want to think about

00:01:54.983 --> 00:01:58.260
is the circumference of
the top of the shell.

00:02:00.920 --> 00:02:04.090
We know circumference
is 2 pi times radius.

00:02:04.090 --> 00:02:08.150
We just need to know what
the radius of the shell is.

00:02:08.150 --> 00:02:10.840
What is that
distance going to be?

00:02:10.840 --> 00:02:12.730
Well, it's the
horizontal distance

00:02:12.730 --> 00:02:19.010
between x equals 2 and whatever
the x value is right over here.

00:02:19.010 --> 00:02:22.700
So it's going to be
2 minus our x value.

00:02:22.700 --> 00:02:25.720
So this radius, this
distance right over here,

00:02:25.720 --> 00:02:29.050
is going to be 2 minus x.

00:02:29.050 --> 00:02:32.990
And so the circumference is
going to be that times 2 pi.

00:02:32.990 --> 00:02:35.340
2 pi r gives us the
circumference of that circle.

00:02:35.340 --> 00:02:39.440
So 2 pi times 2 minus x.

00:02:39.440 --> 00:02:41.160
And then if we want
the surface area

00:02:41.160 --> 00:02:44.310
of the outside of our
shell, so the area

00:02:44.310 --> 00:02:48.410
is going to be the circumference
2 pi times 2 minus x

00:02:48.410 --> 00:02:50.680
times the height of each shell.

00:02:50.680 --> 00:02:55.100
Now, what is the
height of each shell?

00:02:55.100 --> 00:02:57.250
It's going to be the
vertical distance expressed

00:02:57.250 --> 00:02:58.710
as functions of y.

00:02:58.710 --> 00:03:00.480
So it's going to
be the top boundary

00:03:00.480 --> 00:03:02.960
is y is equal to square root
of x, the bottom boundary is

00:03:02.960 --> 00:03:04.170
y is equal to x squared.

00:03:04.170 --> 00:03:07.715
So it's going to be square
root of x minus x squared.

00:03:11.890 --> 00:03:13.500
Let me do this in the yellow.

00:03:13.500 --> 00:03:20.370
So it's going to be square
root of x minus x squared.

00:03:20.370 --> 00:03:24.410
And so if you want the
volume of a given shell--

00:03:24.410 --> 00:03:26.330
I'll write all this
in white-- it's

00:03:26.330 --> 00:03:30.700
going to be 2 pi
times 2 minus x times

00:03:30.700 --> 00:03:34.220
square root of x
minus x squared.

00:03:34.220 --> 00:03:37.945
So this whole expression,
I just rewrote it,

00:03:37.945 --> 00:03:41.190
is the area, the outside surface
area, of one of these shells.

00:03:41.190 --> 00:03:43.690
If we want the volume, we have
to get a little bit of depth,

00:03:43.690 --> 00:03:47.514
multiply by how deep the
shell is, so times dx.

00:03:47.514 --> 00:03:49.430
And if we want the volume
of this whole thing,

00:03:49.430 --> 00:03:51.020
we just have to
solve all the shells

00:03:51.020 --> 00:03:52.630
for all of the x's
in this interval

00:03:52.630 --> 00:03:56.100
and take the limit as the
dx's get smaller and smaller

00:03:56.100 --> 00:03:57.980
and we have more
and more shells.

00:03:57.980 --> 00:03:59.420
And so, what's our interval?

00:03:59.420 --> 00:04:04.210
Well our x's are going
to go between 0 and 1.

00:04:04.210 --> 00:04:11.360
So that right over there is
the volume of this figure.


Shell method for rotating around horizontal line

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.835
What we're going to do in this
video is take the function y

00:00:03.835 --> 00:00:07.460
is equal to the cube
root of x and then rotate

00:00:07.460 --> 00:00:09.970
this around the x-axis.

00:00:09.970 --> 00:00:11.986
And if we do that, we
get a solid of revolution

00:00:11.986 --> 00:00:12.860
that looks like that.

00:00:12.860 --> 00:00:16.880
And we're doing it between x is
equal to 0 and x is equal to 8.

00:00:16.880 --> 00:00:19.030
And you get something
that looks like this.

00:00:19.030 --> 00:00:21.190
And you could find the
volume of this actually

00:00:21.190 --> 00:00:23.162
quite easily using
the disk method.

00:00:23.162 --> 00:00:25.620
But just to show you that you
could do it an alternate way,

00:00:25.620 --> 00:00:27.530
we're going to use
the shell method.

00:00:27.530 --> 00:00:29.321
But we're going to use
the shell method now

00:00:29.321 --> 00:00:32.630
to rotate around a horizontal
line, and specifically

00:00:32.630 --> 00:00:34.510
the x-axis.

00:00:34.510 --> 00:00:36.660
So how would we do this?

00:00:36.660 --> 00:00:39.360
Well, what we want to do,
what you could imagine,

00:00:39.360 --> 00:00:45.360
is constructing a rectangle
that looks like this.

00:00:45.360 --> 00:00:47.580
Let me do it in
this salmon color.

00:00:47.580 --> 00:00:51.070
So you have a rectangle that
looks something like that.

00:00:51.070 --> 00:00:54.650
Its depth or its height,
you could say, is dy.

00:00:54.650 --> 00:00:58.180
And then its length
right over here

00:00:58.180 --> 00:01:02.780
is going to be 8 minus
whatever x value this is.

00:01:02.780 --> 00:01:03.780
Let me make this clear.

00:01:07.210 --> 00:01:11.580
The width is going to
be 8 minus whatever

00:01:11.580 --> 00:01:14.119
x value this is right over here.

00:01:14.119 --> 00:01:16.660
And you might already realize
that if this is going to be dy,

00:01:16.660 --> 00:01:18.826
we're going to be integrating
with respect to y over

00:01:18.826 --> 00:01:21.420
an interval of y.

00:01:21.420 --> 00:01:24.530
And so we really want to have
everything in terms of y.

00:01:24.530 --> 00:01:28.520
So this x value, whatever
it is, as a function of y.

00:01:28.520 --> 00:01:30.580
So if y is equal to
the cube root of x,

00:01:30.580 --> 00:01:32.320
we can cube both
sides, and we can

00:01:32.320 --> 00:01:36.410
get x is equal to y
to the third power.

00:01:36.410 --> 00:01:40.540
These are equivalent
statements right over here.

00:01:40.540 --> 00:01:45.110
And so, this distance right over
here is going to be 8 minus y

00:01:45.110 --> 00:01:47.980
to the third, if we were to
express it in terms of y.

00:01:47.980 --> 00:01:51.005
And then when you rotate
that thing around the x-axis,

00:01:51.005 --> 00:01:53.910
it's going to construct
the outside of the cylinder

00:01:53.910 --> 00:01:55.370
or shell, as we like to call it.

00:01:55.370 --> 00:01:58.720
And I'll try my best
to draw that shell.

00:01:58.720 --> 00:02:01.560
So it will look
something like this.

00:02:05.980 --> 00:02:13.250
So you're going to have a
shell that looks like this.

00:02:13.250 --> 00:02:14.180
And there you go.

00:02:14.180 --> 00:02:16.510
And actually, this shares
a common boundary right

00:02:16.510 --> 00:02:17.610
over here.

00:02:17.610 --> 00:02:20.350
So you're going to have a shell
that looks something like that.

00:02:20.350 --> 00:02:22.080
So that hopefully
helps a little bit.

00:02:22.080 --> 00:02:23.163
Let me give it some depth.

00:02:25.195 --> 00:02:27.320
You'll have a shell that
looks something like that.

00:02:27.320 --> 00:02:29.500
If we could figure out
the volume of that shell,

00:02:29.500 --> 00:02:34.600
which is really going to be
the area of the outer surface

00:02:34.600 --> 00:02:36.290
area times the depth.

00:02:36.290 --> 00:02:39.090
And then if we were to sum
up all of those shells,

00:02:39.090 --> 00:02:41.730
this shell is for a
particular y in our interval.

00:02:41.730 --> 00:02:44.730
If we were to sum up over all
of the y's in our interval,

00:02:44.730 --> 00:02:46.560
all the volumes of
the shells, then we

00:02:46.560 --> 00:02:48.770
have the volume of this figure.

00:02:48.770 --> 00:02:52.330
So once again, how do we figure
out the volume of a shell?

00:02:52.330 --> 00:02:56.280
Well, we can figure
out the circumference

00:02:56.280 --> 00:03:00.900
of I guess the left or the
right, in this case-- the left

00:03:00.900 --> 00:03:02.774
or the right of our cylinder.

00:03:02.774 --> 00:03:04.440
If we can figure out
that circumference,

00:03:04.440 --> 00:03:10.720
that circumference is equal
to 2 pi times the radius.

00:03:10.720 --> 00:03:12.360
And what is the radius?

00:03:12.360 --> 00:03:15.990
The radius of these things are
just going to be your y value.

00:03:15.990 --> 00:03:18.020
That's this distance
right over here.

00:03:18.020 --> 00:03:19.580
That's just going
to be the y value.

00:03:19.580 --> 00:03:24.590
So it's going to be
equal to 2 pi times y.

00:03:24.590 --> 00:03:28.040
And then if we want the
area of the outer surface,

00:03:28.040 --> 00:03:30.300
we just multiply
the circumference

00:03:30.300 --> 00:03:34.260
times the width of our cylinder.

00:03:36.960 --> 00:03:38.300
So let me write it over here.

00:03:38.300 --> 00:03:46.720
So outer surface area is
going to be equal to our 2 pi

00:03:46.720 --> 00:03:52.675
y times our 8 minus
y to the third.

00:03:56.020 --> 00:03:57.840
8 minus y to the third
is just this length.

00:03:57.840 --> 00:03:59.423
You multiply that
times circumference.

00:03:59.423 --> 00:04:01.180
You get the outer surface area.

00:04:01.180 --> 00:04:05.370
Now, if you want to find the
volume of this one shell,

00:04:05.370 --> 00:04:07.830
it's going to be the
outer surface area.

00:04:07.830 --> 00:04:12.780
2 pi y times 8 minus y to
the third times the depth,

00:04:12.780 --> 00:04:17.130
times this dy right over here.

00:04:17.130 --> 00:04:18.290
I'll do the dy in purple.

00:04:23.030 --> 00:04:24.860
So that's the
volume of one shell.

00:04:24.860 --> 00:04:28.440
If we want to find the volume of
the entire solid of revolution,

00:04:28.440 --> 00:04:31.132
we have to sum all these up
and then take the limit as they

00:04:31.132 --> 00:04:32.590
become infinitely
thin, and we have

00:04:32.590 --> 00:04:34.550
an infinite number
of these shells.

00:04:34.550 --> 00:04:37.690
So we're going to take the
sum from, and remember,

00:04:37.690 --> 00:04:38.790
we're dealing in y.

00:04:38.790 --> 00:04:42.220
So the volume is
going to be equal to,

00:04:42.220 --> 00:04:44.170
so what's our interval
in terms of y?

00:04:44.170 --> 00:04:47.900
So y definitely starts off at
0, and when x is equal to 8,

00:04:47.900 --> 00:04:49.040
what is y?

00:04:49.040 --> 00:04:52.280
Well, 8 to the 1/3
power is just 2.

00:04:52.280 --> 00:04:55.020
So y is 2, this value
right over here.

00:04:55.020 --> 00:04:56.520
Let me make it a
little bit clearer.

00:04:56.520 --> 00:04:58.140
This value right over here is 2.

00:04:58.140 --> 00:05:00.700
So y goes from 0 to 2, and
we've set up our integral.

00:05:00.700 --> 00:05:02.450
And this one looks
pretty straightforward.

00:05:02.450 --> 00:05:04.491
So I think we can crank
through it in this video.

00:05:04.491 --> 00:05:08.140
So this is going to be equal
to, we can take out the 2 pi,

00:05:08.140 --> 00:05:13.630
2 pi times the definite
integral from 0 to 2.

00:05:13.630 --> 00:05:18.910
Let's multiply the y of
8y minus y to the fourth.

00:05:18.910 --> 00:05:20.285
All of that dy.

00:05:23.060 --> 00:05:28.680
This is going to be equal to
2 pi times the anti-derivative

00:05:28.680 --> 00:05:30.960
of this business.

00:05:30.960 --> 00:05:34.510
Anti-derivative of
8y is 4y squared.

00:05:37.080 --> 00:05:41.310
Anti-derivative of 1
negative y to the fourth

00:05:41.310 --> 00:05:44.960
is negative y to
the fifth over 5.

00:05:44.960 --> 00:05:47.910
And we're going to
evaluate it at 0 and 2.

00:05:47.910 --> 00:05:51.455
So this is going to be
equal to 2 pi times,

00:05:51.455 --> 00:05:56.884
we evaluate this business at 2,
2 squared is 4 times 4 is 16.

00:05:56.884 --> 00:06:03.320
16 minus 2 to the fifth is 32.

00:06:03.320 --> 00:06:04.200
So minus 32/5.

00:06:07.260 --> 00:06:10.100
And then, you evaluate this
stuff at 0, you just get the 0.

00:06:10.100 --> 00:06:11.980
So that's what we're left with.

00:06:11.980 --> 00:06:13.800
And now, we just
have to simplify

00:06:13.800 --> 00:06:15.080
this thing a little bit.

00:06:15.080 --> 00:06:15.720
So let's see.

00:06:15.720 --> 00:06:18.630
16 over 5, this part
right over here.

00:06:22.480 --> 00:06:25.650
16 is the same thing,
I should say, as 80/5.

00:06:25.650 --> 00:06:27.580
And from that, we
are subtracting 32/5.

00:06:31.860 --> 00:06:35.150
And so, that is equal to 48/5.

00:06:35.150 --> 00:06:38.787
So all of this business
is equal to 48/5.

00:06:38.787 --> 00:06:39.620
Did I do that right?

00:06:39.620 --> 00:06:43.470
80 minus 30 is 50, and then
minus another 2 gets us to 48.

00:06:43.470 --> 00:06:45.880
So this is 48/5 times 2 pi.

00:06:45.880 --> 00:06:47.920
And now we deserve
our drum roll.

00:06:47.920 --> 00:06:50.100
48 times 2 is 96.

00:06:50.100 --> 00:06:51.610
Let me do this in
a new color, just

00:06:51.610 --> 00:06:53.720
to emphasize that
we're at the end.

00:06:53.720 --> 00:06:58.502
48 times 2 is 96
times pi over 5.

00:06:58.502 --> 00:07:00.210
So once again, this
is something that you

00:07:00.210 --> 00:07:03.870
could have solved using the
disk method in terms of x.

00:07:03.870 --> 00:07:06.380
And we're just showing that
you could also solve it

00:07:06.380 --> 00:07:09.410
in the shell method
in terms of y.

00:07:09.410 --> 00:07:11.710
The shell or the hollowed
out cylinder method,

00:07:11.710 --> 00:07:14.730
whatever you want to
call it, in terms of y.


Evaluating integral for shell method example

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.990
Where we left off
in the last video,

00:00:01.990 --> 00:00:03.400
we had set up a
definite integral

00:00:03.400 --> 00:00:07.810
using the shell method for this
strange solid of revolution.

00:00:07.810 --> 00:00:09.780
So now, let's just
evaluate the integral.

00:00:09.780 --> 00:00:12.760
And like we've seen many times
in these type of problems,

00:00:12.760 --> 00:00:16.160
we really just have to do
some polynomial multiplication

00:00:16.160 --> 00:00:16.870
right over here.

00:00:16.870 --> 00:00:19.640
So x minus 3 squared, well,
that's pretty straightforward.

00:00:19.640 --> 00:00:24.569
That's going to be x
squared minus 6x plus 9.

00:00:24.569 --> 00:00:26.610
And we're going to multiply
that times x minus 1.

00:00:26.610 --> 00:00:27.568
So let's do that first.

00:00:27.568 --> 00:00:30.530
So multiply that
times x minus 1.

00:00:30.530 --> 00:00:33.790
And so negative 1
times 9 is negative 9.

00:00:33.790 --> 00:00:39.370
Negative 1 times negative 6
is positive 6, positive 6x.

00:00:39.370 --> 00:00:44.980
Negative 1 times x squared
is negative x squared.

00:00:44.980 --> 00:00:48.198
Now x times 9 is 9x.

00:00:48.198 --> 00:00:53.220
x times negative 6x is
negative 6x squared.

00:00:53.220 --> 00:00:56.760
And then x times x
squared is x to the third.

00:00:56.760 --> 00:01:01.660
And so we get x
to the third minus

00:01:01.660 --> 00:01:09.090
7x squared plus 15x minus 9.

00:01:09.090 --> 00:01:12.020
So we just multiplied x minus
3 squared times x minus 1,

00:01:12.020 --> 00:01:14.374
and then we have to
multiply that times x.

00:01:14.374 --> 00:01:16.040
So we could essentially
raise the degree

00:01:16.040 --> 00:01:17.040
of each of these things.

00:01:20.500 --> 00:01:23.400
At least it's easier now
take the anti-derivative.

00:01:23.400 --> 00:01:28.210
It's equal to 2 pi times the
definite integral from 1 to 3

00:01:28.210 --> 00:01:30.450
of this stuff times x.

00:01:30.450 --> 00:01:36.840
So it's going to be
x to the fourth minus

00:01:36.840 --> 00:01:44.630
7x to the third power
plus 15x squared minus 9x.

00:01:44.630 --> 00:01:46.012
And then, of course, dx.

00:01:46.012 --> 00:01:50.720
And I'll make the dx in
that same nice blue color.

00:01:50.720 --> 00:01:52.980
Now let's just take
the anti-derivative.

00:01:52.980 --> 00:01:57.040
So this is going to be equal to
2 pi times the anti-derivative

00:01:57.040 --> 00:01:59.400
of all of this business.

00:01:59.400 --> 00:02:01.960
We're going to evaluate
it at 3 and subtract it

00:02:01.960 --> 00:02:03.990
when it's evaluated at 1.

00:02:03.990 --> 00:02:06.980
So the anti-derivative
of x to the fourth

00:02:06.980 --> 00:02:08.427
is x to the fifth over 5.

00:02:12.090 --> 00:02:14.880
The anti-derivative
of x to the third

00:02:14.880 --> 00:02:16.480
is x to the fourth
over 4, and we're

00:02:16.480 --> 00:02:18.700
going to multiply
that times negative 7.

00:02:18.700 --> 00:02:24.850
So it's negative 7x
to the fourth over 4.

00:02:24.850 --> 00:02:31.610
And then the anti-derivative
of 15 x squared,

00:02:31.610 --> 00:02:34.780
that's going to be 15 times
x to the third over 3.

00:02:34.780 --> 00:02:40.320
15 divided by 3 is 5, so
it's plus 5x to the third.

00:02:40.320 --> 00:02:43.490
And then finally,
the anti-derivative

00:02:43.490 --> 00:02:52.999
of negative 9x, that's going to
be negative 9x squared over 2.

00:02:52.999 --> 00:02:53.790
And you can verify.

00:02:53.790 --> 00:02:55.248
If you take the
derivative of this,

00:02:55.248 --> 00:02:56.990
you get this business
right over here.

00:02:56.990 --> 00:03:03.610
And so this is going
to be equal to 2 pi.

00:03:03.610 --> 00:03:09.660
And so let's evaluate all
of this business at 3.

00:03:09.660 --> 00:03:11.600
So when you evaluate
it at 3, you

00:03:11.600 --> 00:03:16.060
have 3 to the
fifth power over 5.

00:03:16.060 --> 00:03:19.700
And I believe 3 to the fifth
is 243, but I'll verify.

00:03:19.700 --> 00:03:24.580
3 to the third is 27, 3 to
the fourth is equal to 81,

00:03:24.580 --> 00:03:26.665
3 to the fifth is 243.

00:03:26.665 --> 00:03:29.040
So this is going to give us
some hairy math to deal with.

00:03:29.040 --> 00:03:34.420
So it's going to be 243 over 5.

00:03:34.420 --> 00:03:36.150
3 to the fourth
power, that's 81.

00:03:36.150 --> 00:03:38.900
But then we have to
multiply 81 times 7.

00:03:41.920 --> 00:03:45.250
So we're going to get 567.

00:03:45.250 --> 00:03:46.690
Is that right?

00:03:46.690 --> 00:03:49.220
81 times 7.

00:03:49.220 --> 00:03:58.300
7 times 1 is 7, 7 times 8 is 56,
so we're going to get minus 567

00:03:58.300 --> 00:03:58.870
over 4.

00:03:58.870 --> 00:04:01.251
This is going to be really
painful to do the arithmetic

00:04:01.251 --> 00:04:01.750
part.

00:04:01.750 --> 00:04:04.080
But we'll power through it.

00:04:04.080 --> 00:04:09.220
And then we have 5x
to the third is 27.

00:04:09.220 --> 00:04:11.367
27 times 5 is what?

00:04:11.367 --> 00:04:11.866
135?

00:04:13.994 --> 00:04:15.410
I don't want to
make any mistakes.

00:04:15.410 --> 00:04:18.370
27 times 5.

00:04:18.370 --> 00:04:21.650
7 times 5 is 35,
2 times 5 is 10.

00:04:21.650 --> 00:04:22.150
Yep.

00:04:22.150 --> 00:04:22.970
135.

00:04:22.970 --> 00:04:28.720
So plus 135.

00:04:28.720 --> 00:04:32.540
And then finally, we
have minus 9x squared.

00:04:32.540 --> 00:04:35.500
So x squared is 9 times 9 is 81.

00:04:35.500 --> 00:04:38.270
So minus 81 over 2.

00:04:38.270 --> 00:04:41.590
So that's all of this
business evaluated at 3.

00:04:41.590 --> 00:04:43.560
And from that, we're
going to subtract it

00:04:43.560 --> 00:04:45.555
when it's evaluated at 1.

00:04:51.230 --> 00:04:52.500
Let's do this.

00:04:52.500 --> 00:05:10.426
So we get 1/5 minus 7/4
plus 5, and then minus 9/2.

00:05:15.210 --> 00:05:18.050
And what we are left with is
just a really hairy fractions

00:05:18.050 --> 00:05:18.550
problem.

00:05:18.550 --> 00:05:21.790
So I will just hope that I
don't make a careless mistake

00:05:21.790 --> 00:05:22.670
at this point.

00:05:22.670 --> 00:05:23.670
So let's try to do this.

00:05:23.670 --> 00:05:26.890
This is going to be
equal to 2 pi times,

00:05:26.890 --> 00:05:29.710
and if we wanted to find
a common multiple here,

00:05:29.710 --> 00:05:32.310
it looks like it
would have to be 20.

00:05:32.310 --> 00:05:36.330
Least common multiple
of 5 and 4 and 2 is 20.

00:05:36.330 --> 00:05:41.310
So this is going to give us
243 over 5 is the same thing.

00:05:41.310 --> 00:05:48.720
243 times 4 is going to give
us 3 times 4 is 12, 4 times 4

00:05:48.720 --> 00:05:51.450
is 16, plus 1 is 17.

00:05:51.450 --> 00:05:56.150
2 times 4 is 8, plus 1 is 9.

00:05:56.150 --> 00:06:02.410
So we have 972 over 20.

00:06:02.410 --> 00:06:05.775
And then we have to
multiply 567 times 5.

00:06:09.900 --> 00:06:12.690
So you can see the arithmetic
is the most painful part here.

00:06:12.690 --> 00:06:21.960
7 times 5 is 35, 6 times 5 is
30, plus 3 is 33, 5 times 5

00:06:21.960 --> 00:06:24.350
is 25, plus 3 is 28.

00:06:24.350 --> 00:06:32.710
So we have 2,835 over
4, and then 135 over 20.

00:06:32.710 --> 00:06:39.470
Well, 135 times 2 is going to be
270, and then times another 10

00:06:39.470 --> 00:06:41.120
is 2,700.

00:06:41.120 --> 00:06:45.547
So plus 2,700 over 20.

00:06:45.547 --> 00:06:46.380
Did I do that right?

00:06:46.380 --> 00:06:47.510
Yeah, that's right.

00:06:47.510 --> 00:06:51.100
And then finally, 81 over 2.

00:06:51.100 --> 00:06:54.830
That's going to be the
same thing as negative 810.

00:06:54.830 --> 00:06:56.700
Let me do that same color.

00:06:56.700 --> 00:07:03.250
Negative 810 over 20.

00:07:03.250 --> 00:07:05.960
Numerator and denominator
both multiplied by 10.

00:07:05.960 --> 00:07:07.080
And then let's see.

00:07:07.080 --> 00:07:11.870
Negative 1/5, that's the
same thing as negative 4/20.

00:07:15.290 --> 00:07:17.950
It's going to be positive
7/4 is the same thing

00:07:17.950 --> 00:07:21.400
as positive 35/20.

00:07:21.400 --> 00:07:24.620
And it's going to
be a negative 5

00:07:24.620 --> 00:07:31.240
is the same thing as
negative 100 over 20.

00:07:31.240 --> 00:07:39.754
And then finally, it's
going to be a positive.

00:07:39.754 --> 00:07:41.420
I don't want make
that careless mistake.

00:07:41.420 --> 00:07:42.200
I want to make sure I
get the signs right.

00:07:42.200 --> 00:07:43.350
After I distribute
this negative,

00:07:43.350 --> 00:07:45.075
it's going to be a
positive 9/2, which

00:07:45.075 --> 00:07:49.902
is the same thing
as plus 90 over 20.

00:07:49.902 --> 00:07:51.110
Did I do all the signs right?

00:07:51.110 --> 00:07:56.270
Negative 1/5, positive for
this one, so positive 7/4,

00:07:56.270 --> 00:08:00.500
negative 5, and then
positive 90 over 20.

00:08:00.500 --> 00:08:03.930
And so now, I just have to
do some hard core addition.

00:08:03.930 --> 00:08:06.030
So let's do it.

00:08:06.030 --> 00:08:08.530
So first, I'll take
all of the positives

00:08:08.530 --> 00:08:11.210
and then I'll subtract out the
negatives, just to simplify it,

00:08:11.210 --> 00:08:13.270
so I have to minimize
the number of times.

00:08:13.270 --> 00:08:15.020
Well, I'll add all the
positives together,

00:08:15.020 --> 00:08:16.895
and then I'll add all
the negatives together.

00:08:16.895 --> 00:08:19.410
And that ought to make it
one subtraction problem.

00:08:19.410 --> 00:08:26.570
So 972 plus 2,700
plus 35 plus 90.

00:08:26.570 --> 00:08:27.910
So let me just write it down.

00:08:27.910 --> 00:08:32.870
So this is 2,700 plus 972.

00:08:32.870 --> 00:08:35.720
I should probably take out
a calculator at this point,

00:08:35.720 --> 00:08:38.000
but I'll just do it by
hand, since I've already

00:08:38.000 --> 00:08:39.580
done so much of it by hand.

00:08:39.580 --> 00:08:45.100
Plus 972 plus 90 plus 35.

00:08:48.250 --> 00:08:50.301
So we get a 7.

00:08:50.301 --> 00:08:54.860
7 plus 9 is 16, plus 3 is 19.

00:08:54.860 --> 00:08:55.830
Did I do that right?

00:08:55.830 --> 00:09:01.500
Yeah, 16 plus 3 is 19, and
this is 17, and this is a 3.

00:09:01.500 --> 00:09:05.430
So we have 3,797 when we add
in all the positive numerators.

00:09:05.430 --> 00:09:07.990
And then all the negative
numerators, let's see.

00:09:07.990 --> 00:09:10.115
I'm going to add them
together to see how negative.

00:09:10.115 --> 00:09:15.840
So 2,835, 810, 4, and 100.

00:09:15.840 --> 00:09:28.030
So if I add 2,835, 810, let
me see, 100, and 4, this

00:09:28.030 --> 00:09:30.340
is how much negative I
have to subtract from that.

00:09:30.340 --> 00:09:37.750
5 plus 4 is 9, 3 plus 1 is
4, 8 plus 8 plus 1 is 17,

00:09:37.750 --> 00:09:39.880
and then you have a 3.

00:09:39.880 --> 00:09:46.810
So we're going to
subtract 3,749 from 3,797.

00:09:46.810 --> 00:09:49.050
And so this actually
works out quite well.

00:09:49.050 --> 00:09:50.613
That gets us to 48.

00:09:55.995 --> 00:09:58.120
Let me make sure I haven't
made a careless mistake.

00:09:58.120 --> 00:10:01.602
2,835, 810, 100, and 4.

00:10:01.602 --> 00:10:03.310
Those are all the
things I'm subtracting.

00:10:03.310 --> 00:10:08.921
2,700, 972, 90, and 35 are
all the things I added.

00:10:08.921 --> 00:10:09.420
Yep.

00:10:09.420 --> 00:10:16.080
3,797 minus 3,749 is
going to be equal to 48.

00:10:16.080 --> 00:10:19.610
So this whole expression--
we deserve a drum roll now--

00:10:19.610 --> 00:10:27.620
is going to be equal to
2 pi times 48 over 20.

00:10:27.620 --> 00:10:31.730
And both 48 and 20
are divisible by 4.

00:10:31.730 --> 00:10:37.280
So you get 12 over 5.

00:10:37.280 --> 00:10:38.990
My brain is turning
into mush now.

00:10:38.990 --> 00:10:42.690
I'm becoming paranoid that I've
been making careless mistakes.

00:10:42.690 --> 00:10:43.550
We're almost there.

00:10:43.550 --> 00:10:45.890
So it becomes 12 over 5.

00:10:45.890 --> 00:10:52.230
And so our final answer,
12 times 2 is 24.

00:10:52.230 --> 00:10:56.420
So it becomes 24 pi over 5.

00:10:56.420 --> 00:11:00.480
And we are done.


Shell method for rotating around vertical line

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.720
I've got the function y is
equal to x minus 3 squared

00:00:03.720 --> 00:00:05.370
times x minus 1.

00:00:05.370 --> 00:00:08.390
And what I want to do is
think about rotating the part

00:00:08.390 --> 00:00:10.480
of this function that
sits right over here

00:00:10.480 --> 00:00:13.130
between x is equal
to 1 and x equals 3.

00:00:13.130 --> 00:00:15.400
And x equals 3 and x
equals 1 are clearly

00:00:15.400 --> 00:00:18.530
the zeroes of this
function right over here.

00:00:18.530 --> 00:00:22.760
And I want to take this
region and rotate it

00:00:22.760 --> 00:00:25.120
around the y-axis.

00:00:25.120 --> 00:00:27.460
And if I did that,
I'd get a shape

00:00:27.460 --> 00:00:28.760
that looks something like that.

00:00:28.760 --> 00:00:31.880
And I want to figure out
the volume of that shape.

00:00:31.880 --> 00:00:34.480
And what we're going
to do is a new method

00:00:34.480 --> 00:00:37.487
called the shell method.

00:00:37.487 --> 00:00:39.820
And the reason we're going
to use the shell method-- you

00:00:39.820 --> 00:00:41.236
might say, hey,
in the past, we've

00:00:41.236 --> 00:00:43.450
rotated things around
a vertical line before.

00:00:43.450 --> 00:00:44.550
We used the disk method.

00:00:44.550 --> 00:00:46.445
We wrote everything
as a function of y,

00:00:46.445 --> 00:00:47.320
et cetera, et cetera.

00:00:47.320 --> 00:00:48.360
We created all of these disks.

00:00:48.360 --> 00:00:50.440
We figured out the volume
of each of those disks.

00:00:50.440 --> 00:00:52.710
But the problem here is
this is hard to express

00:00:52.710 --> 00:00:55.980
as a function of y.

00:00:55.980 --> 00:00:59.830
How do you solve explicitly
for y right over here?

00:00:59.830 --> 00:01:02.040
So instead, we're going to
keep things in terms of x

00:01:02.040 --> 00:01:05.129
and have a different geometric
visualization for how

00:01:05.129 --> 00:01:06.746
we can come up with the volume.

00:01:06.746 --> 00:01:08.620
What we're going to
imagine instead-- instead

00:01:08.620 --> 00:01:11.820
of constructing disks, we're
going to construct shells.

00:01:11.820 --> 00:01:13.780
And what do I mean by a shell?

00:01:13.780 --> 00:01:19.770
So for each x at the interval,
on this kind of cut of it,

00:01:19.770 --> 00:01:22.400
we can construct a rectangle.

00:01:22.400 --> 00:01:25.010
And what happens if we were
to rotate this rectangle?

00:01:25.010 --> 00:01:29.030
So this is the rectangle
right over here.

00:01:29.030 --> 00:01:33.460
What happens if we rotate this
rectangle around the y-axis

00:01:33.460 --> 00:01:36.740
along with everything else?

00:01:36.740 --> 00:01:38.430
I'll try my best
attempt to draw it.

00:01:38.430 --> 00:01:43.480
It's going to look
something like this.

00:01:43.480 --> 00:01:45.130
This is challenging
my art skills,

00:01:45.130 --> 00:01:46.750
but I think I can handle it.

00:01:46.750 --> 00:01:50.260
So it's going to look
something not too

00:01:50.260 --> 00:01:52.420
dissimilar to that
right over there.

00:01:52.420 --> 00:01:54.930
So it looks like a
hollowed-out cylinder.

00:01:54.930 --> 00:01:56.720
I guess that's why
we call it a shell.

00:01:56.720 --> 00:01:58.200
And it's going to
have some depth.

00:01:58.200 --> 00:01:59.525
The depth is going to be dx.

00:02:08.294 --> 00:02:09.669
And the height
right over here is

00:02:09.669 --> 00:02:11.950
going to be the
value of my function.

00:02:11.950 --> 00:02:12.960
The height is f of x.

00:02:12.960 --> 00:02:17.030
In this case, f of x is x minus
3 squared times x minus 1.

00:02:17.030 --> 00:02:21.620
How do we figure out the
volume of a cylinder like this?

00:02:21.620 --> 00:02:28.570
Well, if we can figure out the
circumference of the cylinder,

00:02:28.570 --> 00:02:31.870
and then multiply that
circumference times the height

00:02:31.870 --> 00:02:33.970
of the cylinder,
we'd essentially

00:02:33.970 --> 00:02:36.890
figure out the area
of the outside surface

00:02:36.890 --> 00:02:38.220
of our cylinder.

00:02:38.220 --> 00:02:40.730
And then if we multiply the
area of the outside surface

00:02:40.730 --> 00:02:43.660
of our cylinder by that
infinitesimally small depth,

00:02:43.660 --> 00:02:45.080
then that'll give
us the volume--

00:02:45.080 --> 00:02:47.714
I shouldn't say
cylinder-- of our shell.

00:02:47.714 --> 00:02:48.630
So let's try to do it.

00:02:48.630 --> 00:02:51.630
What is the
circumference of a shell?

00:02:55.560 --> 00:02:58.150
What is the circumference
of one shell going to be?

00:02:58.150 --> 00:03:02.950
Well, it's going to be 2 pi
times the radius of that shell.

00:03:02.950 --> 00:03:05.130
We need to express this
as a function of x.

00:03:05.130 --> 00:03:06.810
And so what is that going to be?

00:03:06.810 --> 00:03:08.720
It's going to be 2 pi.

00:03:08.720 --> 00:03:11.210
So for a given x,
what is the radius?

00:03:11.210 --> 00:03:15.150
Well, the radius right over here
is just the horizontal distance

00:03:15.150 --> 00:03:18.540
between the y-axis and that x.

00:03:18.540 --> 00:03:20.120
So that's just x.

00:03:20.120 --> 00:03:21.590
So the circumference,
in this case,

00:03:21.590 --> 00:03:26.560
is just going to
be 2 pi times x.

00:03:26.560 --> 00:03:33.580
Now, what is the height going to
be for any one of those shells?

00:03:33.580 --> 00:03:37.100
The height is
going to be f of x.

00:03:37.100 --> 00:03:38.710
This is right over here.

00:03:38.710 --> 00:03:42.920
And so what is going to be the
surface area of the outside?

00:03:42.920 --> 00:03:47.830
So let me put this in quotes--
"outside" surface area.

00:03:50.690 --> 00:03:53.007
I'm not worried about the
depth right now, the dx.

00:03:53.007 --> 00:03:55.340
I'm not worried about this
top part and the bottom part.

00:03:55.340 --> 00:03:58.237
I'm just worried about
the outside surface area.

00:03:58.237 --> 00:03:59.820
Well, the outside
surface area is just

00:03:59.820 --> 00:04:01.778
going to be the circumference
times the height.

00:04:01.778 --> 00:04:07.670
It's going to be 2
pi x times f of x.

00:04:07.670 --> 00:04:09.516
And in this situation,
in the situation

00:04:09.516 --> 00:04:10.890
we're looking at
right over here,

00:04:10.890 --> 00:04:17.920
that's going to be 2 pi x
times x minus 3 squared times x

00:04:17.920 --> 00:04:19.630
minus 1.

00:04:19.630 --> 00:04:21.250
Now, what's going
to be the volume?

00:04:21.250 --> 00:04:26.690
So the volume of the
shell-- shell volume--

00:04:26.690 --> 00:04:31.680
is just going to be all
this business times dx.

00:04:31.680 --> 00:04:38.670
So it's going to be 2
pi x times f of x dx.

00:04:38.670 --> 00:04:41.850
And so now we're ready to
integrate over the interval.

00:04:41.850 --> 00:04:46.190
So the volume of
our entire shape

00:04:46.190 --> 00:04:48.020
is going to be the
definite integral.

00:04:48.020 --> 00:04:50.436
We're going to integrate over
all the x's in the interval,

00:04:50.436 --> 00:04:58.480
from x is equal to 1 to x
is equal to 3 of this thing.

00:04:58.480 --> 00:05:00.220
And we could take
the 2 pi out front.

00:05:00.220 --> 00:05:02.730
So we'll put 2 pi out front.

00:05:02.730 --> 00:05:05.970
And on the inside,
we have x times f

00:05:05.970 --> 00:05:09.910
of x, which in our
situation, is this business.

00:05:09.910 --> 00:05:17.000
So it's going to be x times
x minus 3 squared times x

00:05:17.000 --> 00:05:18.700
minus 1.

00:05:18.700 --> 00:05:22.715
And then, of course,
we have our dx.

00:05:22.715 --> 00:05:23.590
So there you have it.

00:05:23.590 --> 00:05:26.740
Using the shell method, we have
set up our definite integral

00:05:26.740 --> 00:05:31.320
for the volume of this
strange-looking shape right

00:05:31.320 --> 00:05:33.020
over there.


Washer method rotating around vertical line (not y-axis), part 1

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.400
What we're going to
do in this video is

00:00:02.400 --> 00:00:05.090
take the region
between the two curves,

00:00:05.090 --> 00:00:08.119
y is equal to square
root of x on top and y

00:00:08.119 --> 00:00:10.240
is equal to x
squared on the bottom

00:00:10.240 --> 00:00:13.780
and rotate it around a vertical
line that is not the y-axis.

00:00:13.780 --> 00:00:16.030
So we're going to rotate it
around the vertical line x

00:00:16.030 --> 00:00:16.690
is equal to 2.

00:00:16.690 --> 00:00:19.310
We're going to rotate it
right around like that.

00:00:19.310 --> 00:00:21.990
And if we rotate
it like that, we

00:00:21.990 --> 00:00:25.150
will get this shape, so
this strange-looking shape.

00:00:25.150 --> 00:00:26.500
It's hollowed out in the middle.

00:00:26.500 --> 00:00:29.680
You can kind of use the y equals
x squared part kind of hollows

00:00:29.680 --> 00:00:30.690
out the middle.

00:00:30.690 --> 00:00:32.630
And it's really the
stuff in between

00:00:32.630 --> 00:00:36.810
that forms the wall
of this rotated shape.

00:00:36.810 --> 00:00:40.080
So let's think about how we
can figure out the volume.

00:00:40.080 --> 00:00:42.524
And we're going to do it using
the disk method, sometimes

00:00:42.524 --> 00:00:43.940
called the ring
method-- actually,

00:00:43.940 --> 00:00:45.773
it's going to be more
of the washer method--

00:00:45.773 --> 00:00:47.760
to do this one right over here.

00:00:47.760 --> 00:00:50.070
So we're rotating
around a vertical line.

00:00:50.070 --> 00:00:53.210
We want to use our disk
or ring or washer method.

00:00:53.210 --> 00:00:55.290
And so it'll be
really helpful to have

00:00:55.290 --> 00:00:58.770
a bunch of rings stacked
up in the y direction.

00:00:58.770 --> 00:01:01.530
So we're probably going to want
to integrate with respect to y.

00:01:01.530 --> 00:01:03.560
Let me make it clear
what I'm talking about.

00:01:03.560 --> 00:01:06.330
Each of our rings are going
to look something like this.

00:01:06.330 --> 00:01:08.280
So let me do my best
attempt to draw.

00:01:08.280 --> 00:01:13.150
So that's the inner
radius of the ring defined

00:01:13.150 --> 00:01:17.120
by y is equal to x squared.

00:01:17.120 --> 00:01:19.110
And then the outside
radius of the ring

00:01:19.110 --> 00:01:20.500
might look something like this.

00:01:25.090 --> 00:01:27.660
My best attempt to
draw it reasonably.

00:01:27.660 --> 00:01:34.590
The outside radius of the ring
might look something like that.

00:01:34.590 --> 00:01:39.950
And the depth of the ring would
be a little dy just like that.

00:01:39.950 --> 00:01:42.440
So let me draw the depth.

00:01:42.440 --> 00:01:43.944
So let me see, right over here.

00:01:43.944 --> 00:01:46.110
You could kind of view it
as the height of the ring,

00:01:46.110 --> 00:01:48.550
because we are thinking
in a vertical direction.

00:01:48.550 --> 00:01:50.050
And that would be
a ring for a given

00:01:50.050 --> 00:01:53.840
y for, say, this
y right over here.

00:01:53.840 --> 00:01:55.840
It's essentially
the ring generated

00:01:55.840 --> 00:02:02.100
if you were to take this
rectangle of height dy

00:02:02.100 --> 00:02:05.012
and rotate it around
the line x equals 2.

00:02:05.012 --> 00:02:06.470
And we're going to
construct a ring

00:02:06.470 --> 00:02:12.730
like that for each of
the y's in our interval.

00:02:12.730 --> 00:02:15.540
So you could imagine stacking
up a whole set of rings just

00:02:15.540 --> 00:02:18.710
like this and then taking
the sum of the volumes of all

00:02:18.710 --> 00:02:21.160
of those rings and
the limit of that side

00:02:21.160 --> 00:02:24.336
as you have an infinite number
of rings with-- or close

00:02:24.336 --> 00:02:25.710
to infinite-- or
really, infinite

00:02:25.710 --> 00:02:28.470
number of rings with
infinitesmally small height

00:02:28.470 --> 00:02:30.130
or depth or dy.

00:02:30.130 --> 00:02:32.470
So let's figure out
how we would do this.

00:02:32.470 --> 00:02:34.235
Well, we know that
all we have to do

00:02:34.235 --> 00:02:36.360
is figure out what the
volume of one of these rings

00:02:36.360 --> 00:02:38.940
are for given y
as a function of y

00:02:38.940 --> 00:02:41.825
and then integrate along all
of them, sum them all up.

00:02:41.825 --> 00:02:44.200
So let's figure out what the
volume of one of these rings

00:02:44.200 --> 00:02:44.980
are.

00:02:44.980 --> 00:02:46.480
And to do it, we're
going to express

00:02:46.480 --> 00:02:48.990
these functions
as functions of y.

00:02:48.990 --> 00:02:53.350
So our purple function, y is
equal to square root of x,

00:02:53.350 --> 00:03:00.040
if we square both sides,
we would get y squared

00:03:00.040 --> 00:03:00.695
is equal to x.

00:03:00.695 --> 00:03:03.070
And let me swap the sides so
it makes it clear that now x

00:03:03.070 --> 00:03:04.462
is a function of y.

00:03:04.462 --> 00:03:07.600
x is equal to y squared.

00:03:07.600 --> 00:03:09.300
That's our top
function the way we've

00:03:09.300 --> 00:03:12.350
drawn a kind of outer
shell for our figure.

00:03:12.350 --> 00:03:16.800
And then y is
equal to x squared.

00:03:16.800 --> 00:03:19.642
If you take the principal root
of both sides of that and it

00:03:19.642 --> 00:03:22.100
all works out because we're
operating in the first quadrant

00:03:22.100 --> 00:03:22.600
here.

00:03:22.600 --> 00:03:24.380
That's the part of it
that we care about.

00:03:24.380 --> 00:03:28.590
So you're going to get x is
equal to the square root of y.

00:03:28.590 --> 00:03:31.310
That is our yellow
function right over there.

00:03:31.310 --> 00:03:33.630
Now, how do we
figure out the area

00:03:33.630 --> 00:03:38.470
of the surface of one of these
rings or one of these washers?

00:03:38.470 --> 00:03:40.982
Well, the area-- let
me do this in orange

00:03:40.982 --> 00:03:42.440
because I drew that
ring in orange.

00:03:42.440 --> 00:03:45.100
So the area of the
surface right over here

00:03:45.100 --> 00:03:51.240
in orange as a
function of y is going

00:03:51.240 --> 00:03:54.130
to be equal to the
area of the circle

00:03:54.130 --> 00:03:55.810
if I just consider
the outer radius,

00:03:55.810 --> 00:03:58.370
and then I subtract out the
area of the circle constructed

00:03:58.370 --> 00:04:01.210
by the inner radius, just
kind of subtract it out.

00:04:01.210 --> 00:04:03.010
So the outer-circle
radius, so it's

00:04:03.010 --> 00:04:14.560
going to be pi times outer
radius squared minus pi-- let

00:04:14.560 --> 00:04:18.042
me write it this
way-- outer-- well,

00:04:18.042 --> 00:04:19.500
I'll just write
it-- outer radius--

00:04:19.500 --> 00:04:22.830
this is going to be as a
function of y-- outer radius

00:04:22.830 --> 00:04:30.490
squared minus pi times
inner radius squared.

00:04:30.490 --> 00:04:32.680
And we want this all
to be a function of y.

00:04:32.680 --> 00:04:38.610
So the outer radius
as a function of y

00:04:38.610 --> 00:04:40.149
is going to be what?

00:04:40.149 --> 00:04:42.190
Well, it might be easier
to visualize-- actually,

00:04:42.190 --> 00:04:43.190
I'll try it both places.

00:04:43.190 --> 00:04:47.570
So it's this entire
distance right over here,

00:04:47.570 --> 00:04:51.550
essentially, the distance
between the vertical line,

00:04:51.550 --> 00:04:53.940
the horizontal distance
between our vertical line,

00:04:53.940 --> 00:04:55.035
and our outer function.

00:05:02.160 --> 00:05:03.970
So if you think about
it in terms of x,

00:05:03.970 --> 00:05:06.370
It's going to be
2 minus whatever

00:05:06.370 --> 00:05:08.970
the x value is right over here.

00:05:08.970 --> 00:05:12.540
So the x value right over
here is going to be y squared.

00:05:12.540 --> 00:05:14.610
Remember, we want this
is a function of y.

00:05:14.610 --> 00:05:16.580
So our outer radius,
this whole distance,

00:05:16.580 --> 00:05:20.480
is going to be 2 minus the x
value here as a function of y.

00:05:20.480 --> 00:05:22.410
That x value is y squared.

00:05:22.410 --> 00:05:26.770
So the outer radius
is 2 minus x-- sorry.

00:05:26.770 --> 00:05:28.350
2 minus y squared.

00:05:28.350 --> 00:05:31.390
We want it as a function of y.

00:05:31.390 --> 00:05:37.774
And then our inner radius is
going to be equal to what?

00:05:37.774 --> 00:05:39.440
Well, that's going
to be the difference,

00:05:39.440 --> 00:05:42.570
the horizontal distance,
between this vertical line

00:05:42.570 --> 00:05:45.950
and our inner function,
our inner boundary.

00:05:45.950 --> 00:05:48.440
So it's going to be the
horizontal distance between two

00:05:48.440 --> 00:05:50.420
and whatever x value this is.

00:05:50.420 --> 00:05:55.140
But this x value as a function
of y is just square root of y.

00:05:55.140 --> 00:06:00.200
So it's going to be 2
minus square root of y.

00:06:00.200 --> 00:06:02.960
And so now, we can come up
with an expression for area.

00:06:02.960 --> 00:06:05.350
It's going to be-- and I'll
just factor out-- actually,

00:06:05.350 --> 00:06:06.350
I'll leave the pi there.

00:06:06.350 --> 00:06:08.730
So it's going to be
pi-- right over here--

00:06:08.730 --> 00:06:12.070
it's going to be pi times
outer radius squared.

00:06:12.070 --> 00:06:14.500
Well, the outer
radius is 2 minus y

00:06:14.500 --> 00:06:18.153
squared-- and let me just--
well, I'll just write it.

00:06:18.153 --> 00:06:20.890
2 minus y squared--
and we're going

00:06:20.890 --> 00:06:26.740
to square that-- squared,
minus pi times the inner radius

00:06:26.740 --> 00:06:27.330
squared.

00:06:27.330 --> 00:06:28.760
Well, we already
figured that out.

00:06:28.760 --> 00:06:32.240
The inner radius is 2
minus square root of y.

00:06:32.240 --> 00:06:34.340
And we're going to
square that one, too.

00:06:34.340 --> 00:06:37.330
So this gives us the
area of one of our rings

00:06:37.330 --> 00:06:41.670
as a function of y, the
top of the ring, where

00:06:41.670 --> 00:06:42.712
I shaded in orange.

00:06:42.712 --> 00:06:44.920
And now, if we want the
volume of one of those rings,

00:06:44.920 --> 00:06:47.294
we have to multiply it by its
depth or its height the way

00:06:47.294 --> 00:06:49.070
we've drawn it right over here.

00:06:49.070 --> 00:06:52.030
And its height-- we've done this
multiple times already right

00:06:52.030 --> 00:06:55.580
over here-- is an
infinitesimal change in y.

00:06:55.580 --> 00:06:59.840
So we're going to multiply
all that business times dy.

00:06:59.840 --> 00:07:01.660
This is the volume
of one of our rings.

00:07:01.660 --> 00:07:04.631
And then we want to sum up all
of the rings over our interval.

00:07:04.631 --> 00:07:07.130
So we're going to sum up all
of the rings over the interval.

00:07:07.130 --> 00:07:08.630
And when you take
the integral sign,

00:07:08.630 --> 00:07:11.260
it's a sum where you're
taking the limit as you

00:07:11.260 --> 00:07:14.690
have an infinite number of rings
that become infinitesimally

00:07:14.690 --> 00:07:17.970
small in height or depth,
depending on how you view it.

00:07:17.970 --> 00:07:19.880
And what's our interval?

00:07:19.880 --> 00:07:21.550
So we've looked at
this multiple times.

00:07:21.550 --> 00:07:23.180
These two graphs-- you
could do it by inspection.

00:07:23.180 --> 00:07:24.763
You could try to
solve it in some way,

00:07:24.763 --> 00:07:27.190
but it's pretty obvious
that they intersect at--

00:07:27.190 --> 00:07:29.150
remember we care
about our y interval.

00:07:29.150 --> 00:07:34.355
They intersect at y is equal
to 0 and y is equal to 1.

00:07:38.404 --> 00:07:39.320
And there you have it.

00:07:39.320 --> 00:07:42.900
We've set up our
integral for the volume

00:07:42.900 --> 00:07:45.106
of this shape right over here.

00:07:45.106 --> 00:07:46.980
I'll leave you there,
and in this next video,

00:07:46.980 --> 00:07:50.120
we will just evaluate
this integral.


Calculating integral disc around vertical line

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.520
In the last video, we had
set up the definite integral

00:00:03.520 --> 00:00:07.060
to evaluate the volume of
this upside-down gumdrop

00:00:07.060 --> 00:00:08.159
truffle-looking thing.

00:00:08.159 --> 00:00:09.700
So now in this video,
we can actually

00:00:09.700 --> 00:00:11.310
evaluate the definite integral.

00:00:11.310 --> 00:00:12.810
So what we need to
do is really just

00:00:12.810 --> 00:00:17.132
expand out this expression, this
square root of y plus 1 plus 2.

00:00:17.132 --> 00:00:17.840
So let's do that.

00:00:17.840 --> 00:00:21.440
So this is going to be equal to
pi times the definite integral

00:00:21.440 --> 00:00:24.200
from y is equal to negative
1 to y is equal to 3.

00:00:24.200 --> 00:00:26.210
If you expand this out,
you get square root of y

00:00:26.210 --> 00:00:30.005
plus 1 squared, which is
just going to be y plus 1.

00:00:30.005 --> 00:00:32.380
And then you're going to have
2 times the product of both

00:00:32.380 --> 00:00:33.005
of these terms.

00:00:33.005 --> 00:00:34.910
2 times 2 times square
root of y plus 1

00:00:34.910 --> 00:00:39.940
is going to be plus 4 times
the square root of y plus 1.

00:00:39.940 --> 00:00:43.660
And then you have 2
squared, so plus 4.

00:00:43.660 --> 00:00:46.170
So you have this
whole thing times dy.

00:00:46.170 --> 00:00:47.440
We can simplify a little bit.

00:00:47.440 --> 00:00:48.990
You have a 1 plus a 4.

00:00:48.990 --> 00:00:52.322
We can add the 1 to
the 4 and get a 5.

00:00:52.322 --> 00:00:54.280
And now we're ready to
take the antiderivative.

00:00:54.280 --> 00:00:58.430
So this is going to be
equal to pi times-- let's

00:00:58.430 --> 00:01:03.790
take the antiderivative of all
of this business-- pi times--

00:01:03.790 --> 00:01:05.129
and I'll color code it.

00:01:05.129 --> 00:01:10.250
The antiderivative of y
is just y squared over 2.

00:01:10.250 --> 00:01:12.880
The antiderivative of
4 times the square root

00:01:12.880 --> 00:01:14.400
of y plus 1-- you
just really have

00:01:14.400 --> 00:01:19.090
to think of it as 4 times
y plus 1 to the 1/2 power.

00:01:19.090 --> 00:01:20.720
We could use u
substitution explicitly,

00:01:20.720 --> 00:01:22.595
but you probably are
pretty practiced in this

00:01:22.595 --> 00:01:23.803
and can do this in your head.

00:01:23.803 --> 00:01:25.590
You have y plus 1
raised to the 1/2 power.

00:01:25.590 --> 00:01:27.220
Derivative of y
plus 1 is just 1,

00:01:27.220 --> 00:01:28.740
which is essentially out here.

00:01:28.740 --> 00:01:30.810
So if you did u
substitution, you

00:01:30.810 --> 00:01:33.770
would say u is
equal to y plus 1.

00:01:33.770 --> 00:01:38.020
But this antiderivative is
going to be equal to-- well,

00:01:38.020 --> 00:01:39.740
if you increment
this exponent, you

00:01:39.740 --> 00:01:43.430
get 3/2 multiplied by
the reciprocal 2/3.

00:01:43.430 --> 00:01:45.490
2/3 times 4 is 8/3.

00:01:45.490 --> 00:01:51.560
So it's plus 8/3 times
y plus 1 to the 3/2.

00:01:51.560 --> 00:01:52.602
And you can verify.

00:01:52.602 --> 00:01:54.060
If you take the
derivative here you

00:01:54.060 --> 00:01:56.340
will get this expression
right over here.

00:01:56.340 --> 00:01:58.550
3/2 times 8/3 is 4.

00:01:58.550 --> 00:02:01.670
Decremented, you have y
plus 1 to the 1/2 power.

00:02:01.670 --> 00:02:04.650
And then finally,
you have-- let's see.

00:02:04.650 --> 00:02:06.580
What color have I not used yet?

00:02:06.580 --> 00:02:07.880
Finally, you have this 5.

00:02:07.880 --> 00:02:11.850
The antiderivative
of 5 is just 5y.

00:02:11.850 --> 00:02:17.840
And we are going to evaluate
it at 3 and at negative 1,

00:02:17.840 --> 00:02:20.260
y equals 3 and y
equals negative 1.

00:02:20.260 --> 00:02:22.420
So this is going
to be equal to pi.

00:02:22.420 --> 00:02:25.490
So let's evaluate all
this business at 3.

00:02:25.490 --> 00:02:30.450
So 3 squared over 2 is 9/2.

00:02:30.450 --> 00:02:33.384
3 plus 1 is 4 to the 3/2.

00:02:33.384 --> 00:02:34.550
Well, that's-- so let's see.

00:02:34.550 --> 00:02:38.210
If square root of 4 is 2 to the
third power is 8, 8 times 8/3

00:02:38.210 --> 00:02:42.730
is 64/3, so plus 64/3.

00:02:42.730 --> 00:02:44.740
You have 5 times 3.

00:02:44.740 --> 00:02:48.550
Well, that's going
to be 15-- plus 15.

00:02:48.550 --> 00:02:50.570
And from that, we're
going to subtract

00:02:50.570 --> 00:02:53.310
all this business
evaluated at negative 1.

00:02:53.310 --> 00:02:55.370
So you have negative
1 squared over 2.

00:02:55.370 --> 00:02:57.450
Well, that's just 1/2.

00:02:57.450 --> 00:03:00.920
Negative 1 plus 1 is
0 to the 3/2 power.

00:03:00.920 --> 00:03:02.440
That's going to be 0 times 8/3.

00:03:02.440 --> 00:03:04.981
This is all going to be 0, so
we don't have to even write it.

00:03:04.981 --> 00:03:08.900
And then finally, you
have negative 1 times 5.

00:03:08.900 --> 00:03:12.220
Well, that's just
going to be negative 5.

00:03:12.220 --> 00:03:13.860
And we are in the home stretch.

00:03:13.860 --> 00:03:16.470
We really just have to do
a little bit of arithmetic,

00:03:16.470 --> 00:03:19.370
add some hairy fractions
right over here.

00:03:19.370 --> 00:03:20.760
So let's do it.

00:03:20.760 --> 00:03:25.060
So this whole thing is going
to simplify to pi times--

00:03:25.060 --> 00:03:26.310
and it looks like-- let's see.

00:03:26.310 --> 00:03:28.560
Our least common multiple
of all of these denominators

00:03:28.560 --> 00:03:29.650
is going to be 6.

00:03:29.650 --> 00:03:32.150
So let's put everything
over a denominator of 6.

00:03:32.150 --> 00:03:36.910
So 9/2 is the same
thing as 27/6.

00:03:36.910 --> 00:03:40.093
64/3 is the same thing as 128/6.

00:03:42.656 --> 00:03:47.010
15 is the same thing as 90/6.

00:03:50.410 --> 00:03:53.320
1/2 is the same thing as 3/6.

00:03:53.320 --> 00:03:55.380
So you would distribute
the negative sign.

00:03:55.380 --> 00:03:58.230
So this is negative 3/6.

00:03:58.230 --> 00:04:00.770
And negative times
negative is positive.

00:04:00.770 --> 00:04:07.860
5 is the same thing
as 30/6, so plus 30/6.

00:04:07.860 --> 00:04:09.990
And so this is going to
give us-- our denominator

00:04:09.990 --> 00:04:11.570
is going to be over 6.

00:04:11.570 --> 00:04:14.590
We're going to multiply
something times pi.

00:04:14.590 --> 00:04:15.760
We have this pi over here.

00:04:15.760 --> 00:04:18.176
And then we just have to figure
out what the numerator is.

00:04:18.176 --> 00:04:22.290
So let's see if I can
do this in my head.

00:04:22.290 --> 00:04:31.620
So 27 plus 128 is going
to be-- let me see.

00:04:31.620 --> 00:04:35.270
That's going to be 140, 155?

00:04:35.270 --> 00:04:36.420
Is that right?

00:04:36.420 --> 00:04:38.980
155?

00:04:38.980 --> 00:04:43.000
Let's see, if we get to 48
plus another 7-- yeah, 155.

00:04:43.000 --> 00:04:47.957
Plus 90 gets us to 245.

00:04:47.957 --> 00:04:48.540
Is that right?

00:04:48.540 --> 00:04:49.040
Yeah.

00:04:49.040 --> 00:04:50.830
Plus 90 gets us to 245.

00:04:50.830 --> 00:04:55.740
Minus 3, you subtract 3
from that, you get to 242.

00:04:55.740 --> 00:05:01.400
And then you add 30 to
that, you get to 272.

00:05:01.400 --> 00:05:05.424
So we're left with 272 pi/6.

00:05:05.424 --> 00:05:06.590
But then we can-- let's see.

00:05:06.590 --> 00:05:10.210
272 and 6 are both
divisible by 2.

00:05:10.210 --> 00:05:12.030
So this is equal to-- let's see.

00:05:12.030 --> 00:05:19.287
272 divided by 2 is
going to be 136 pi over--

00:05:19.287 --> 00:05:21.370
and if you divide this
denominator right over here

00:05:21.370 --> 00:05:23.467
by 2-- over 3.

00:05:23.467 --> 00:05:24.050
Is that right?

00:05:24.050 --> 00:05:24.550
Yeah.

00:05:24.550 --> 00:05:26.240
136 pi/3.

00:05:26.240 --> 00:05:29.580
And 136 is not divisible by 3.

00:05:29.580 --> 00:05:32.120
So we have it as
simplified as we can.

00:05:32.120 --> 00:05:34.490
This right over
here is the volume

00:05:34.490 --> 00:05:38.900
of our little upside-down
gumdrop-looking thing.


Disc method rotating around vertical line

WEBVTT
Kind: captions
Language: en

00:00:00.555 --> 00:00:02.430
Let's do another example,
and this time we're

00:00:02.430 --> 00:00:06.080
going to rotate our function
around a vertical line that

00:00:06.080 --> 00:00:06.994
is not the y-axis.

00:00:06.994 --> 00:00:08.410
And if we do that--
so we're going

00:00:08.410 --> 00:00:11.300
to rotate y is equal
to x squared minus 1--

00:00:11.300 --> 00:00:12.800
or at least this
part of it-- we're

00:00:12.800 --> 00:00:14.670
going to rotate it
around the vertical line

00:00:14.670 --> 00:00:16.660
x is equal to negative 2.

00:00:16.660 --> 00:00:18.520
And if we do that, we
get this gumball shape

00:00:18.520 --> 00:00:19.880
that looks something like this.

00:00:19.880 --> 00:00:21.463
So what I want to
do is I want to find

00:00:21.463 --> 00:00:23.840
the volume of this
using the disc method.

00:00:23.840 --> 00:00:27.730
So what I want to do is
construct some discs.

00:00:27.730 --> 00:00:30.860
So that's one of the
discs right over here.

00:00:30.860 --> 00:00:33.240
It's going to have some
depth, and that depth

00:00:33.240 --> 00:00:39.230
is going to be dy
right over there.

00:00:39.230 --> 00:00:42.930
And it's going to have
some area on top of it

00:00:42.930 --> 00:00:47.210
that is a function of
any given y that I have.

00:00:47.210 --> 00:00:48.950
So the volume of a
given disc is going

00:00:48.950 --> 00:00:51.660
to be the area as
a function of y

00:00:51.660 --> 00:00:54.399
times the depth of
the disc times dy.

00:00:54.399 --> 00:00:56.690
And then we just have to
integrate it over the interval

00:00:56.690 --> 00:00:59.340
that we care about, and we're
doing it all in terms of y.

00:00:59.340 --> 00:01:01.090
And in this case, we're
going to integrate

00:01:01.090 --> 00:01:03.892
from y is equal
to-- well, this is

00:01:03.892 --> 00:01:05.850
going to hit-- this
y-intercept right over here

00:01:05.850 --> 00:01:07.480
is y is equal to negative 1.

00:01:07.480 --> 00:01:10.610
And let's go all the
way to y is equal to,

00:01:10.610 --> 00:01:15.470
let's say y is equal
to 3 right over here.

00:01:15.470 --> 00:01:17.580
So from y equal negative
1 to y equals 3.

00:01:22.170 --> 00:01:25.080
And that's going to
give us the volume

00:01:25.080 --> 00:01:29.010
of our upside-down
gumdrop-type-looking thing.

00:01:29.010 --> 00:01:30.880
So the key here, so
that we can start

00:01:30.880 --> 00:01:33.520
evaluating the double
integral, is to just figure out

00:01:33.520 --> 00:01:37.520
what the area of each of these
discs are as a function of y.

00:01:37.520 --> 00:01:40.960
And we know that area
as a function of y

00:01:40.960 --> 00:01:45.760
is just going to be pi times
radius as a function of y

00:01:45.760 --> 00:01:46.550
squared.

00:01:46.550 --> 00:01:49.860
So the real key is, what is
the radius as a function of y

00:01:49.860 --> 00:01:51.200
for any one of these y's?

00:01:51.200 --> 00:01:56.486
So what is the radius
as a function of y?

00:01:56.486 --> 00:01:58.110
So let's think about
that a little bit.

00:01:58.110 --> 00:01:59.510
What is this curve?

00:01:59.510 --> 00:02:01.226
Well, let's write it
as a function of y.

00:02:01.226 --> 00:02:03.600
If you add 1 to both sides--
and I'm going to swap sides,

00:02:03.600 --> 00:02:06.980
so you'll get x squared
is equal to y plus 1.

00:02:06.980 --> 00:02:10.180
I just added 1 to both sides
and then swapped sides.

00:02:10.180 --> 00:02:12.980
And you get x is equal
to the principal root

00:02:12.980 --> 00:02:15.662
of the square root of y plus 1.

00:02:15.662 --> 00:02:20.120
So this we can write as x-- or
we can even write it as f of y

00:02:20.120 --> 00:02:25.780
if we want-- f of y is equal
to the square root of y plus 1.

00:02:25.780 --> 00:02:28.940
Or we could say x is equal
to a function of y, which

00:02:28.940 --> 00:02:30.726
is the square root of y plus 1.

00:02:30.726 --> 00:02:34.380
So what's the distance
here at any point?

00:02:34.380 --> 00:02:38.640
Well, this distance-- let
me make it very clear.

00:02:38.640 --> 00:02:40.730
So it's going to be
our total distance

00:02:40.730 --> 00:02:43.020
in the horizontal direction.

00:02:43.020 --> 00:02:45.810
So this first part as we're--
and I'm going to do it

00:02:45.810 --> 00:02:47.270
in a different
color so we can see.

00:02:47.270 --> 00:02:49.600
So this part right
over here is just

00:02:49.600 --> 00:02:51.454
going to be the value
of the function.

00:02:51.454 --> 00:02:52.870
It's going to give
you an x value.

00:02:52.870 --> 00:02:56.950
But then you have to add another
2 to go all the way over here.

00:02:56.950 --> 00:03:02.080
So your entire radius
as a function of y

00:03:02.080 --> 00:03:05.875
is going to be equal to the
square root of y plus 1.

00:03:05.875 --> 00:03:08.620
This essentially will give
you one of these x values when

00:03:08.620 --> 00:03:11.430
you're sitting on this curve,
this x as a function of y,

00:03:11.430 --> 00:03:13.100
it'll give you one
of these x values.

00:03:13.100 --> 00:03:17.534
And then from that, you
add another 2, so plus 2.

00:03:17.534 --> 00:03:19.950
Another way of thinking about
it, you get an x value here,

00:03:19.950 --> 00:03:24.070
and from that x value
you subtract out x

00:03:24.070 --> 00:03:26.102
is equal to negative 2.

00:03:26.102 --> 00:03:28.060
And when you subtract x
is equal to negative 2,

00:03:28.060 --> 00:03:29.370
you're adding 2 here.

00:03:29.370 --> 00:03:31.230
But hopefully, this
makes intuitive sense.

00:03:31.230 --> 00:03:33.600
This is the x value-- let
me do this in a better

00:03:33.600 --> 00:03:36.940
color-- this right over here,
this distance right over here,

00:03:36.940 --> 00:03:39.280
is the x value you
get when you just

00:03:39.280 --> 00:03:41.595
evaluate the function of y.

00:03:41.595 --> 00:03:43.220
But then if you wanted
the full radius,

00:03:43.220 --> 00:03:45.470
you have to go another
2 to go to the center

00:03:45.470 --> 00:03:47.460
of our axis of rotation.

00:03:47.460 --> 00:03:51.070
Once again, if you just take
a given y right over there,

00:03:51.070 --> 00:03:53.210
you evaluate the y,
you get an x value.

00:03:53.210 --> 00:03:55.480
That x value will just
give you this distance.

00:03:55.480 --> 00:03:58.490
If you want the full distance,
you have to subtract negative 2

00:03:58.490 --> 00:04:00.200
from that x value,
which is essentially

00:04:00.200 --> 00:04:03.530
the same thing as adding
2 to get our full radius.

00:04:03.530 --> 00:04:07.310
So our radius as a function of
y is this thing right over here.

00:04:07.310 --> 00:04:09.380
So substituting
back into this, we

00:04:09.380 --> 00:04:12.190
can now write our definite
integral for our volume.

00:04:12.190 --> 00:04:18.329
The volume is going to be
equal to the definite integral

00:04:18.329 --> 00:04:24.800
from negative 1 to 3 of pi
times our radius squared dy.

00:04:24.800 --> 00:04:26.390
So I can write the
pi out here-- we've

00:04:26.390 --> 00:04:29.420
done this multiple times--
times radius squared--

00:04:29.420 --> 00:04:35.770
so it's going to be square root
of y plus 1 plus 2 squared--

00:04:35.770 --> 00:04:39.830
that's our radius-- times dy.

00:04:39.830 --> 00:04:42.010
So we've set up the
definite integral.

00:04:42.010 --> 00:04:45.130
And now we just have
to evaluate this thing.

00:04:45.130 --> 00:04:47.310
And I'll save that
for the next video.

00:04:47.310 --> 00:04:51.150
And I encourage you to
try this out on your own.


Washer method rotating around horizontal line (not x-axis), part 2

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.029
Where we left off
in the last video,

00:00:02.029 --> 00:00:03.890
we'd actually set up
our definite integral

00:00:03.890 --> 00:00:05.770
to figure out the
volume of this figure.

00:00:05.770 --> 00:00:07.491
So now we just have
to evaluate it.

00:00:07.491 --> 00:00:08.990
And really the
hardest part is going

00:00:08.990 --> 00:00:12.860
to be simplifying that
and that right over there.

00:00:12.860 --> 00:00:14.730
So let's get to it.

00:00:14.730 --> 00:00:15.980
So what is this thing squared?

00:00:15.980 --> 00:00:17.480
Well, looks like
we're going to have

00:00:17.480 --> 00:00:19.700
to do a little bit of
polynomial multiplication here.

00:00:19.700 --> 00:00:21.500
So I'll go into that
same color I had.

00:00:21.500 --> 00:00:27.740
So we're going to have 4 minus x
squared plus 2x times 4 minus--

00:00:27.740 --> 00:00:31.756
actually, let me write that
in the order of the terms,

00:00:31.756 --> 00:00:32.880
or the degree of the terms.

00:00:32.880 --> 00:00:37.240
It's negative x squared
plus 2x plus 4-- I

00:00:37.240 --> 00:00:40.450
just switched the order of
these things-- times negative x

00:00:40.450 --> 00:00:43.615
squared plus 2x plus 4.

00:00:43.615 --> 00:00:47.262
So we're just going to
multiply these two things.

00:00:47.262 --> 00:00:49.220
I shouldn't even write
a multiplication symbol,

00:00:49.220 --> 00:00:50.590
it looks too much like an x.

00:00:50.590 --> 00:00:53.190
So 4 times 4 is 16.

00:00:53.190 --> 00:00:56.490
4 times 2x is 8x.

00:00:56.490 --> 00:01:01.540
4 times negative x squared
is negative 4x squared.

00:01:01.540 --> 00:01:04.550
2x times 4 is 8x.

00:01:04.550 --> 00:01:08.280
2x times 2x is 4x squared.

00:01:08.280 --> 00:01:10.580
And then 2x times
negative x squared

00:01:10.580 --> 00:01:15.032
is negative 2x to
the third power.

00:01:15.032 --> 00:01:17.490
And now we just have to multiply
negative x times all this.

00:01:17.490 --> 00:01:19.050
So negative-- or
negative x squared.

00:01:19.050 --> 00:01:23.830
Negative x squared times
4 is negative 4x squared.

00:01:23.830 --> 00:01:30.180
Negative x squared times 2x is
negative 2x to the third power.

00:01:30.180 --> 00:01:31.730
And then negative
x squared times

00:01:31.730 --> 00:01:34.960
negative x squared is
positive x to the fourth.

00:01:34.960 --> 00:01:37.030
So it's going to be
positive x to the fourth.

00:01:37.030 --> 00:01:40.160
And now we just have to
add up all of these terms.

00:01:40.160 --> 00:01:43.770
And we get, let's see,
we get x to the fourth--

00:01:43.770 --> 00:01:47.400
add these two-- minus
4x to the third.

00:01:47.400 --> 00:01:49.470
And then this cancels
with this, but we still

00:01:49.470 --> 00:01:52.470
have that, so minus 4x squared.

00:01:52.470 --> 00:01:56.150
You add these two right
over here, you get plus 16x.

00:01:56.150 --> 00:01:58.570
And then you have plus 16.

00:01:58.570 --> 00:02:01.090
So that's this expanded out.

00:02:01.090 --> 00:02:05.960
And now if we want to-- 4
minus x times 4 minus x.

00:02:05.960 --> 00:02:09.372
So if we just have 4
minus x times 4 minus x,

00:02:09.372 --> 00:02:11.080
we could actually do
it this way as well.

00:02:11.080 --> 00:02:12.920
But that's just going
to be 4 times 4, which

00:02:12.920 --> 00:02:17.860
is 16, plus 4 times negative
x, which is negative 4x,

00:02:17.860 --> 00:02:20.421
negative x times 4,
another negative 4x,

00:02:20.421 --> 00:02:22.420
and then negative x times
negative x is equal to

00:02:22.420 --> 00:02:23.852
plus x squared.

00:02:23.852 --> 00:02:25.310
So if we were to
swap the order, we

00:02:25.310 --> 00:02:32.500
get x squared minus 8x plus 16.

00:02:32.500 --> 00:02:34.450
But we're going
to subtract this.

00:02:34.450 --> 00:02:37.047
So if you have the
negative sign out there,

00:02:37.047 --> 00:02:38.630
we're going to
subtract this business.

00:02:38.630 --> 00:02:40.310
So let's just do
it right over here

00:02:40.310 --> 00:02:41.890
since we already have it set up.

00:02:41.890 --> 00:02:46.360
So we have this minus this.

00:02:46.360 --> 00:02:49.202
We're going to
subtract this, or we

00:02:49.202 --> 00:02:50.410
could add the negative of it.

00:02:50.410 --> 00:02:58.360
So we'll put negative x
squared plus 8x minus 16.

00:02:58.360 --> 00:03:00.830
So I'm just going to add
the negative of this.

00:03:00.830 --> 00:03:07.970
And we get-- and I'll
do this in a new color--

00:03:07.970 --> 00:03:11.790
let's see, we get x
to the fourth minus 4x

00:03:11.790 --> 00:03:21.420
to the third power minus
5x squared plus 24x,

00:03:21.420 --> 00:03:23.060
and then these cancel out.

00:03:23.060 --> 00:03:24.650
So that's what we are left with.

00:03:24.650 --> 00:03:27.000
And so that's the
inside of our integral.

00:03:27.000 --> 00:03:29.100
So we're going to take the
integral of this thing,

00:03:29.100 --> 00:03:31.183
just so I don't have to
keep rewriting this thing,

00:03:31.183 --> 00:03:34.400
from 0 until, if I
remember correctly, 3.

00:03:34.400 --> 00:03:39.250
Yup, from 0 to 3, of this, dx.

00:03:39.250 --> 00:03:43.290
And then we had a
pi out front here.

00:03:43.290 --> 00:03:45.210
So I'll just take that
out of the integral.

00:03:45.210 --> 00:03:48.770
So times pi.

00:03:48.770 --> 00:03:51.260
Well, now we just to
take the antiderivative.

00:03:51.260 --> 00:03:56.470
And this is going to be equal
to pi times antiderivative

00:03:56.470 --> 00:04:00.250
of x to the fourth is
x to the fifth over 5.

00:04:00.250 --> 00:04:02.170
Antiderivative of
4x to the third

00:04:02.170 --> 00:04:03.740
is actually x to the fourth.

00:04:03.740 --> 00:04:06.570
So this is going to be
minus x to the fourth--

00:04:06.570 --> 00:04:07.480
you can verify that.

00:04:07.480 --> 00:04:10.030
Derivative is 4x, and
then you decrement

00:04:10.030 --> 00:04:11.510
the exponent, 4x to the third.

00:04:11.510 --> 00:04:12.710
So that works out.

00:04:12.710 --> 00:04:17.420
And then the antiderivative
of this is negative 5/3 x

00:04:17.420 --> 00:04:18.050
to the third.

00:04:18.050 --> 00:04:20.920
Just incremented the
exponent and divided by that.

00:04:20.920 --> 00:04:29.370
And then you have plus 24x
squared over 2, or 12x squared.

00:04:29.370 --> 00:04:31.280
And we're going
to evaluate that.

00:04:31.280 --> 00:04:34.380
I like to match colors
for my opening and closing

00:04:34.380 --> 00:04:35.074
parentheses.

00:04:35.074 --> 00:04:36.740
We're going to evaluate
that-- actually,

00:04:36.740 --> 00:04:38.380
let me do it as brackets.

00:04:38.380 --> 00:04:42.690
We're going to evaluate
that from 0 to 3.

00:04:42.690 --> 00:04:45.590
So this is going to be
equal to pi times-- let's

00:04:45.590 --> 00:04:50.770
evaluate all this business at 3.

00:04:50.770 --> 00:04:53.500
So we're going to get
3 to the fifth power.

00:04:56.060 --> 00:05:02.390
So let's see, 3 to the
third is equal to 27.

00:05:02.390 --> 00:05:05.910
3 to the fourth is equal to 81.

00:05:05.910 --> 00:05:13.130
3 to the fifth is going to be
equal to-- this times 3 is 243.

00:05:13.130 --> 00:05:15.640
So it's going to
be 243-- it's going

00:05:15.640 --> 00:05:20.700
to be some hairy math-- 243 over
5 minus-- well, 3 to the fourth

00:05:20.700 --> 00:05:21.670
is 81.

00:05:21.670 --> 00:05:23.530
Minus 81.

00:05:23.530 --> 00:05:29.340
Minus, let's see, we're going
to have 3 to the third times--

00:05:29.340 --> 00:05:33.140
let's see, so it's going
to be minus 5 over 3

00:05:33.140 --> 00:05:35.460
times 3 to the third power.

00:05:35.460 --> 00:05:37.740
So times 27.

00:05:37.740 --> 00:05:41.950
Well, 27 divided by 3 is just 9.

00:05:41.950 --> 00:05:43.864
9 times 5 is 45.

00:05:43.864 --> 00:05:45.030
So let's just simplify that.

00:05:45.030 --> 00:05:48.670
So this is going
to be equal to 45.

00:05:48.670 --> 00:05:49.550
Did I do that right?

00:05:49.550 --> 00:05:50.050
Yeah.

00:05:50.050 --> 00:05:52.654
It's essentially going to
be like 3 squared times 5,

00:05:52.654 --> 00:05:54.070
because you're
dividing by 3 here.

00:05:54.070 --> 00:05:56.350
So it's going to have 45.

00:05:56.350 --> 00:05:58.700
And then finally,
3 squared is 9.

00:05:58.700 --> 00:06:01.800
9 times 12 is 108.

00:06:01.800 --> 00:06:04.240
So plus 108.

00:06:04.240 --> 00:06:06.519
These problems that
involve hairy arithmetic

00:06:06.519 --> 00:06:08.060
are always the most
stressful for me,

00:06:08.060 --> 00:06:09.926
but I'll try not to
get too stressed.

00:06:09.926 --> 00:06:12.550
And then we're going to subtract
out this whole thing evaluated

00:06:12.550 --> 00:06:14.740
at 0, but lucky for us,
that's pretty simple.

00:06:14.740 --> 00:06:17.420
This evaluated is 0, 0, 0, 0, 0.

00:06:17.420 --> 00:06:19.390
So we're going to
subtract out 0,

00:06:19.390 --> 00:06:20.950
which simplifies
things a good bit.

00:06:20.950 --> 00:06:24.980
So now we just have to do some
hairy fraction arithmetic.

00:06:24.980 --> 00:06:26.550
So let's do it.

00:06:26.550 --> 00:06:29.277
So what I'm going to do first
is simplify all of this part.

00:06:29.277 --> 00:06:31.110
And then I'm going to
worry about putting it

00:06:31.110 --> 00:06:34.030
over a denominator of 5.

00:06:34.030 --> 00:06:35.340
So let's see what we have.

00:06:35.340 --> 00:06:38.930
We have negative 81 minus 45.

00:06:38.930 --> 00:06:45.350
So these two right over
here become negative 126.

00:06:45.350 --> 00:06:49.300
And then negative 126 plus 108.

00:06:49.300 --> 00:06:52.120
Well, that's just going to be
the same thing as negative 26

00:06:52.120 --> 00:06:56.230
plus 8, which is going
to be negative 18.

00:06:56.230 --> 00:07:00.340
So this whole thing
simplifies to negative 18.

00:07:00.340 --> 00:07:03.020
Did I do that right?

00:07:03.020 --> 00:07:06.820
So this is negative 126.

00:07:06.820 --> 00:07:09.270
And then negative 126-- yes.

00:07:09.270 --> 00:07:12.480
It will be negative 18.

00:07:12.480 --> 00:07:15.360
So now we just have to
write negative 18 over 5.

00:07:15.360 --> 00:07:21.170
Negative 18 over 5 is the same
thing as negative-- let's see,

00:07:21.170 --> 00:07:24.050
5 times 10 is 50, plus 40.

00:07:24.050 --> 00:07:26.580
So that's going to be
negative 90 over 5.

00:07:26.580 --> 00:07:28.310
So this whole thing
has simplified

00:07:28.310 --> 00:07:37.980
to-- is equal to pi
times 243 over 5 minus 90

00:07:37.980 --> 00:07:44.810
over 5, which is
equal to pi-- guys,

00:07:44.810 --> 00:07:46.010
I deserve a drum roll now.

00:07:46.010 --> 00:07:47.970
This was some hairy mathematics.

00:07:47.970 --> 00:07:55.400
So 243 minus 90 is
going to be 153 over 5.

00:07:55.400 --> 00:07:59.570
Or we can write this
as 153 pi over 5.

00:07:59.570 --> 00:08:01.200
And after all that
math, you sometimes

00:08:01.200 --> 00:08:02.750
forget what we were
doing in the first place.

00:08:02.750 --> 00:08:04.640
We were figuring out
this right over here

00:08:04.640 --> 00:08:10.210
is the volume of
this figure that

00:08:10.210 --> 00:08:13.720
had this little inside
of it bored out.


Washer method rotating around horizontal line (not x-axis), part 1

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.780
Now let's do a really
interesting problem.

00:00:02.780 --> 00:00:05.290
So I have y equals x,
and y is equal to x

00:00:05.290 --> 00:00:07.840
squared minus 2x
right over here.

00:00:07.840 --> 00:00:09.530
And we're going to
rotate the region

00:00:09.530 --> 00:00:11.020
in between these two functions.

00:00:11.020 --> 00:00:13.277
So that's this region
right over here.

00:00:13.277 --> 00:00:15.610
And we're not going to rotate
it just around the x-axis,

00:00:15.610 --> 00:00:19.360
we're going to rotate it around
the horizontal line y equals 4.

00:00:19.360 --> 00:00:21.454
So we're going to
rotate it around this.

00:00:21.454 --> 00:00:23.870
And if we do that, we'll get
a shape that looks like this.

00:00:23.870 --> 00:00:26.550
I drew it ahead of time, just
so I could draw it nicely.

00:00:26.550 --> 00:00:30.210
And as you can see, it looks
like some type of a vase

00:00:30.210 --> 00:00:32.060
with a hole at the bottom.

00:00:32.060 --> 00:00:34.520
And so what we're going to do
is attempt to do this using,

00:00:34.520 --> 00:00:36.395
I guess you'd call it
the washer method which

00:00:36.395 --> 00:00:37.800
is a variant of the disk method.

00:00:37.800 --> 00:00:39.790
So let's construct a washer.

00:00:39.790 --> 00:00:42.350
So let's look at a given x.

00:00:42.350 --> 00:00:45.520
So let's say an x
right over here.

00:00:45.520 --> 00:00:47.750
So let's say that we're
at an x right over there.

00:00:47.750 --> 00:00:48.370
And what we're
going to do is we're

00:00:48.370 --> 00:00:50.160
going to rotate this region.

00:00:50.160 --> 00:00:54.090
We're going to give
it some depth, dx.

00:00:54.090 --> 00:00:55.010
So that is dx.

00:00:55.010 --> 00:00:57.120
We're going to rotate
this around the line y

00:00:57.120 --> 00:00:57.840
is equal to 4.

00:00:57.840 --> 00:01:02.580
So if you were to visualize it
over here, you have some depth.

00:01:02.580 --> 00:01:05.472
And when you rotate it
around, the inner radius

00:01:05.472 --> 00:01:07.680
is going to look like the
inner radius of our washer.

00:01:07.680 --> 00:01:09.440
It's going to look
something like that.

00:01:12.210 --> 00:01:13.900
And then the outer
radius of our washer

00:01:13.900 --> 00:01:17.420
is going to contour
around x squared minus 2x.

00:01:17.420 --> 00:01:21.780
So it's going to
look something--

00:01:21.780 --> 00:01:23.703
my best attempt
to draw it-- it's

00:01:23.703 --> 00:01:25.440
going to look
something like that.

00:01:27.950 --> 00:01:30.810
And of course, our washer
is going to have some depth.

00:01:30.810 --> 00:01:32.250
So let me draw the depth.

00:01:32.250 --> 00:01:35.950
So it's going to
have some depth, dx.

00:01:35.950 --> 00:01:39.940
So this is my best attempt at
drawing some of that the depth.

00:01:39.940 --> 00:01:42.970
So this is the
depth of our washer.

00:01:42.970 --> 00:01:45.440
And then just to make the face
of the washer a little bit

00:01:45.440 --> 00:01:47.230
clearer, let me do it
in this green color.

00:01:47.230 --> 00:01:49.440
So the face of the
washer is going

00:01:49.440 --> 00:01:52.310
to be all of this business.

00:01:52.310 --> 00:01:57.060
All of this business is going
to be the face of our washer.

00:01:57.060 --> 00:01:59.350
So if we can figure
out the volume of one

00:01:59.350 --> 00:02:01.210
of these washers for a
given x, then we just

00:02:01.210 --> 00:02:03.240
have to sum up all of
the washers for all

00:02:03.240 --> 00:02:05.919
of the x's in our interval.

00:02:05.919 --> 00:02:07.710
So let's see if we can
set up the integral,

00:02:07.710 --> 00:02:09.990
and maybe in the
next video we'll

00:02:09.990 --> 00:02:13.820
just forge ahead and actually
evaluate the integral.

00:02:13.820 --> 00:02:15.964
So let's think about the
volume of the washer.

00:02:15.964 --> 00:02:17.630
To think about the
volume of the washer,

00:02:17.630 --> 00:02:19.580
we really just have to
think about the area

00:02:19.580 --> 00:02:21.510
of the face of the washer.

00:02:21.510 --> 00:02:26.510
So area of "face"--
put face in quotes--

00:02:26.510 --> 00:02:28.380
is going to be equal to what?

00:02:28.380 --> 00:02:30.670
Well, it would be the
area of the washer--

00:02:30.670 --> 00:02:32.880
if it wasn't a washer,
if it was just a coin--

00:02:32.880 --> 00:02:35.441
and then subtract out
the area of the part

00:02:35.441 --> 00:02:36.440
that you're cutting out.

00:02:36.440 --> 00:02:38.890
So the area of the
washer if it didn't

00:02:38.890 --> 00:02:40.750
have a hole in the
middle would just

00:02:40.750 --> 00:02:44.285
be pi times the
outer radius squared.

00:02:48.150 --> 00:02:51.090
It would be pi times
this radius squared,

00:02:51.090 --> 00:02:52.950
that we could call
the outer radius.

00:02:52.950 --> 00:02:55.320
And since it's a washer,
we need to subtract out

00:02:55.320 --> 00:02:57.030
the area of this inner circle.

00:02:57.030 --> 00:03:05.500
So minus pi times
inner radius squared.

00:03:05.500 --> 00:03:07.000
So we really just
have to figure out

00:03:07.000 --> 00:03:11.160
what the outer and inner radius,
or radii I should say, are.

00:03:11.160 --> 00:03:12.710
So let's think about it.

00:03:12.710 --> 00:03:19.971
So our outer radius is
going to be equal to what?

00:03:19.971 --> 00:03:21.470
Well, we can visualize
it over here.

00:03:21.470 --> 00:03:23.610
This is our outer
radius, which is also

00:03:23.610 --> 00:03:27.770
going to be equal to
that right over there.

00:03:27.770 --> 00:03:29.720
So that's the distance
between y equals

00:03:29.720 --> 00:03:32.440
4 and the function that's
defining our outside.

00:03:38.250 --> 00:03:40.520
So this is essentially,
this height right over here,

00:03:40.520 --> 00:03:45.280
is going to be equal to 4
minus x squared minus 2x.

00:03:45.280 --> 00:03:48.000
I'm just finding the distance
or the height between these two

00:03:48.000 --> 00:03:48.900
functions.

00:03:48.900 --> 00:03:52.140
So the outer radius is
going to be 4 minus this,

00:03:52.140 --> 00:03:55.100
minus x squared minus
2x, which is just 4

00:03:55.100 --> 00:03:58.520
minus x squared plus 2x.

00:03:58.520 --> 00:03:59.850
Now, what is the inner radius?

00:04:05.080 --> 00:04:06.830
What is that going to be?

00:04:06.830 --> 00:04:11.550
Well, that's just going to
be this distance between y

00:04:11.550 --> 00:04:13.480
equals 4 and y equals x.

00:04:13.480 --> 00:04:15.265
So it's just going
to be 4 minus x.

00:04:19.079 --> 00:04:22.810
So if we wanted to find
the area of the face of one

00:04:22.810 --> 00:04:27.090
of these washers for a
given x, it's going to be--

00:04:27.090 --> 00:04:30.290
and we can factor
out this pi-- it's

00:04:30.290 --> 00:04:34.680
going to be pi times the
outer radius squared,

00:04:34.680 --> 00:04:36.540
which is all of this
business squared.

00:04:36.540 --> 00:04:41.830
So it's going to be 4 minus
x squared plus 2x squared

00:04:41.830 --> 00:04:43.280
minus pi times
the inner radius--

00:04:43.280 --> 00:04:44.780
although we factored
out the pi-- so

00:04:44.780 --> 00:04:46.810
minus the inner radius squared.

00:04:46.810 --> 00:04:51.800
So minus 4 minus x squared.

00:04:51.800 --> 00:04:57.650
So this will give us
the area of the surface

00:04:57.650 --> 00:04:59.280
or the face of one
of these washers.

00:04:59.280 --> 00:05:01.700
If we want the volume of
one of those watchers,

00:05:01.700 --> 00:05:05.040
we then just have to
multiply times the depth, dx.

00:05:08.010 --> 00:05:10.800
And then if we want to actually
find the volume of this entire

00:05:10.800 --> 00:05:14.250
figure, then we just have to
sum up all of these washers

00:05:14.250 --> 00:05:15.950
for each of our x's.

00:05:15.950 --> 00:05:16.830
So let's do that.

00:05:16.830 --> 00:05:19.050
So we're going to sum
up the washers for each

00:05:19.050 --> 00:05:21.300
of our x's and take the
limit as they approach zero,

00:05:21.300 --> 00:05:23.490
but we have to make sure
we got our interval right.

00:05:23.490 --> 00:05:26.260
So what are these-- we care
about the entire region

00:05:26.260 --> 00:05:28.840
between the points
where they intersect.

00:05:28.840 --> 00:05:30.620
So let's make sure
we get our interval.

00:05:30.620 --> 00:05:32.203
So to figure out our
interval, we just

00:05:32.203 --> 00:05:36.070
say when does y equal
x intersect y equal

00:05:36.070 --> 00:05:37.350
x squared minus 2x?

00:05:39.990 --> 00:05:41.630
Let me do this in
a different color.

00:05:41.630 --> 00:05:44.150
We just have to
think about when does

00:05:44.150 --> 00:05:46.245
x equal x squared minus 2x.

00:05:49.370 --> 00:05:51.420
When are our two functions
equal to each other?

00:05:51.420 --> 00:05:53.030
Which is equivalent
to-- if we just

00:05:53.030 --> 00:05:58.800
subtract x from
both sides, we get

00:05:58.800 --> 00:06:02.190
when does x squared
minus 3x equal 0.

00:06:02.190 --> 00:06:05.260
We can factor out an x
on the right hand side.

00:06:05.260 --> 00:06:09.530
So this is going to be when does
x times x minus 3 equal zero.

00:06:09.530 --> 00:06:12.310
Well, if the product is equal
to 0, at least one of these

00:06:12.310 --> 00:06:13.350
need to be equal to 0.

00:06:13.350 --> 00:06:18.380
So x could be equal to 0,
or x minus 3 is equal to 0.

00:06:18.380 --> 00:06:21.280
So x is equal to 0
or x is equal to 3.

00:06:21.280 --> 00:06:24.010
So this is x is 0, and
this right over here

00:06:24.010 --> 00:06:25.850
is x is equal to 3.

00:06:25.850 --> 00:06:27.100
So that gives us our interval.

00:06:27.100 --> 00:06:29.120
We're going to go
from x equals 0

00:06:29.120 --> 00:06:32.850
to x equals 3 to get our volume.

00:06:32.850 --> 00:06:35.000
In the next video,
we'll actually

00:06:35.000 --> 00:06:37.320
evaluate this integral.


Disc method rotation around horizontal line

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.520
Here we've graphed
the function y

00:00:02.520 --> 00:00:04.910
is equal to the
square root of x.

00:00:04.910 --> 00:00:06.910
And we're going to create
a solid of revolution,

00:00:06.910 --> 00:00:08.890
but we're not going to
do it by rotating this

00:00:08.890 --> 00:00:10.990
around the x or the y-axis.

00:00:10.990 --> 00:00:12.830
Instead, we're
going to rotate it

00:00:12.830 --> 00:00:14.920
around another somewhat
arbitrary line.

00:00:14.920 --> 00:00:16.650
And in this case,
I will rotate it

00:00:16.650 --> 00:00:20.140
around the line y is equal to 1.

00:00:20.140 --> 00:00:22.250
So let's say that
this right over here

00:00:22.250 --> 00:00:25.330
is the line y equals 1.

00:00:25.330 --> 00:00:28.915
So this is what we're going to
rotate it around, y equals 1.

00:00:28.915 --> 00:00:31.540
So the first thing we want to do
is visualize what we're doing.

00:00:31.540 --> 00:00:33.350
And we actually care
about the interval.

00:00:33.350 --> 00:00:35.600
So let's say that the
interval is between this point

00:00:35.600 --> 00:00:38.570
right over here where
the two points intersect.

00:00:38.570 --> 00:00:42.830
And let's say between
that and x is equal to 4.

00:00:42.830 --> 00:00:45.810
So let's say that this right
over here, so this is x

00:00:45.810 --> 00:00:46.870
equals 4.

00:00:46.870 --> 00:00:48.830
So it's going to be
just right over here.

00:00:48.830 --> 00:00:50.400
So this is the interval
that we're rotating,

00:00:50.400 --> 00:00:52.358
and we're going to rotate
it around y equals 1,

00:00:52.358 --> 00:00:55.442
not around the x-axis.

00:00:55.442 --> 00:00:56.900
So what would our
figure look like?

00:00:56.900 --> 00:01:02.500
Well, we're going to
rotate it around like this.

00:01:02.500 --> 00:01:07.220
Going to rotate it around
something like this.

00:01:07.220 --> 00:01:08.892
And so your figure
is going to look,

00:01:08.892 --> 00:01:10.600
I guess you could call
it, it will almost

00:01:10.600 --> 00:01:12.280
look like a cone
head, if you view it

00:01:12.280 --> 00:01:14.520
on the side, or
something like a bullet,

00:01:14.520 --> 00:01:16.040
but not quite looking
like a bullet.

00:01:16.040 --> 00:01:21.730
But it would be a shape that
looks something like that.

00:01:21.730 --> 00:01:23.340
So hopefully, we
can visualize this.

00:01:23.340 --> 00:01:25.480
We've done this
several times already,

00:01:25.480 --> 00:01:27.130
attempting to
visualize these shapes.

00:01:27.130 --> 00:01:29.380
But let's think about how
we could actually figure out

00:01:29.380 --> 00:01:32.450
the volume of this
solid of revolution.

00:01:32.450 --> 00:01:34.660
Well, let's just think
about it disk by disk.

00:01:34.660 --> 00:01:37.095
So construct a disk
right over here.

00:01:39.680 --> 00:01:42.500
And we've done this many,
many, many times already

00:01:42.500 --> 00:01:45.029
where we essentially want the
volume of each of these disks.

00:01:45.029 --> 00:01:47.320
And then we're going to take
this for each of these x's

00:01:47.320 --> 00:01:48.861
in our interval,
and then we're going

00:01:48.861 --> 00:01:50.910
to take the sum of the
volumes of all the disks

00:01:50.910 --> 00:01:51.956
to find the volume.

00:01:51.956 --> 00:01:53.330
We're going to
stack them all up,

00:01:53.330 --> 00:01:56.040
or I guess put them next
to each other in this case,

00:01:56.040 --> 00:01:58.710
and find the volume
of our entire figure.

00:01:58.710 --> 00:02:01.060
So to figure out the
volume of each disk,

00:02:01.060 --> 00:02:05.280
we just have to figure
out the area of its face.

00:02:05.280 --> 00:02:06.622
And I'll do that in purple.

00:02:06.622 --> 00:02:08.580
We just have to figure
out the area of the face

00:02:08.580 --> 00:02:11.020
and multiply it by the depth.

00:02:11.020 --> 00:02:12.620
So what's the area of this face?

00:02:12.620 --> 00:02:17.250
Well, it's going to be pi times
the radius of the face squared.

00:02:20.530 --> 00:02:23.150
Now, what's the radius
of the face here?

00:02:23.150 --> 00:02:25.360
Well, it's not just
square root of x.

00:02:25.360 --> 00:02:27.620
Square root of x would
tell us the distance

00:02:27.620 --> 00:02:29.960
between the x-axis
and our function.

00:02:29.960 --> 00:02:32.870
It's now square
root of x minus 1.

00:02:32.870 --> 00:02:37.200
This length right over here.

00:02:37.200 --> 00:02:42.040
The square root of x minus 1
for any given x in our interval.

00:02:42.040 --> 00:02:45.450
So it's going to be equal
to square root of-- let

00:02:45.450 --> 00:02:47.660
me do that same orange
color just to make it clear

00:02:47.660 --> 00:02:48.965
where I'm getting it from.

00:02:48.965 --> 00:02:52.780
It's going to be equal to
square root of x minus 1.

00:02:55.910 --> 00:02:58.460
It's essentially
our function minus

00:02:58.460 --> 00:03:00.790
what we are rotating around.

00:03:00.790 --> 00:03:02.690
That gives us the radius.

00:03:02.690 --> 00:03:05.790
And so this will give us the
area of each of our faces.

00:03:05.790 --> 00:03:09.300
And then we just multiply
that times the depth.

00:03:09.300 --> 00:03:11.259
The depth, of course, is dx.

00:03:11.259 --> 00:03:12.550
We've done that multiple times.

00:03:12.550 --> 00:03:14.470
So times dx.

00:03:14.470 --> 00:03:21.930
This is the thing that we want
to sum up over our interval.

00:03:21.930 --> 00:03:24.660
And our interval-- let's see,
this point right over here

00:03:24.660 --> 00:03:27.460
where the square root
of x is equal to 1,

00:03:27.460 --> 00:03:29.330
that's just going
to be x equals 1.

00:03:29.330 --> 00:03:31.710
This is just x
equals 1 over here.

00:03:31.710 --> 00:03:33.240
And we said that
we would do this

00:03:33.240 --> 00:03:35.172
all the way until x equals 4.

00:03:35.172 --> 00:03:36.880
This was kind of our
end of our interval.

00:03:36.880 --> 00:03:39.920
So this is until x is equal
to-- let me be careful.

00:03:39.920 --> 00:03:43.520
This is the x-axis
right over here.

00:03:43.520 --> 00:03:45.710
All the way until x equals 4.

00:03:45.710 --> 00:03:46.950
So we get confused.

00:03:46.950 --> 00:03:48.620
This is the x-axis.

00:03:48.620 --> 00:03:51.780
So we're going from x
equals 1 to x equals 4.

00:03:51.780 --> 00:03:55.130
And we're essentially
taking this area, one way

00:03:55.130 --> 00:03:58.500
to think about it, between
our square root of x and 1,

00:03:58.500 --> 00:04:00.257
and we're rotating it around 1.

00:04:00.257 --> 00:04:02.340
So we're using these little
disks right over here.

00:04:02.340 --> 00:04:04.980
So the interval between 1 and 4.

00:04:04.980 --> 00:04:07.560
And so this is going to give
us the volume of our solid

00:04:07.560 --> 00:04:08.250
of revolution.

00:04:08.250 --> 00:04:10.430
And so now we just
have to evaluate

00:04:10.430 --> 00:04:12.320
this definite integral.

00:04:12.320 --> 00:04:13.590
So let's give a shot at it.

00:04:13.590 --> 00:04:20.279
So this is going to be equal
to the integral from 1 to 4.

00:04:20.279 --> 00:04:23.450
And we can factor out--
or we could put the pi

00:04:23.450 --> 00:04:24.920
outside the integral sign.

00:04:24.920 --> 00:04:27.030
And then we can expand this out.

00:04:27.030 --> 00:04:29.126
So square root of x squared.

00:04:29.126 --> 00:04:31.950
Now, what we're going to do
is expand this binomial so

00:04:31.950 --> 00:04:35.680
the square root of x minus
1 times the square root of x

00:04:35.680 --> 00:04:36.950
minus 1.

00:04:36.950 --> 00:04:39.870
Square root of x times
square root of x is x.

00:04:39.870 --> 00:04:44.570
Square root of x times negative
1 is negative square root of x.

00:04:44.570 --> 00:04:46.040
Negative 1 times
square root of x

00:04:46.040 --> 00:04:47.930
is another negative
square root of x.

00:04:47.930 --> 00:04:50.080
And then negative
1 times negative 1

00:04:50.080 --> 00:04:52.480
is equal to positive 1.

00:04:52.480 --> 00:04:54.440
So this part right
over here will

00:04:54.440 --> 00:05:00.960
simplify to x minus
2 square roots of x.

00:05:00.960 --> 00:05:03.430
This is minus 2
square roots of x.

00:05:06.570 --> 00:05:08.410
And then you have plus 1.

00:05:08.410 --> 00:05:11.800
And then all of that times dx.

00:05:11.800 --> 00:05:14.660
So this is going to
be equal to-- let's

00:05:14.660 --> 00:05:17.920
put our pi out there--
the antiderivative

00:05:17.920 --> 00:05:19.440
of this business.

00:05:19.440 --> 00:05:25.800
So that the antiderivative
of x is x squared over 2.

00:05:25.800 --> 00:05:29.186
The antiderivative-- so minus
2 times the antiderivative

00:05:29.186 --> 00:05:32.040
of square root of x.

00:05:32.040 --> 00:05:34.830
Well, square root of x is
just x to the 1/2 power,

00:05:34.830 --> 00:05:36.670
so we increment the power by 1.

00:05:36.670 --> 00:05:43.160
So it's going to be x to
the 3/2 power times 2/3.

00:05:43.160 --> 00:05:44.340
So times 2/3.

00:05:44.340 --> 00:05:52.380
Or we could say it's going
to be times 2/3x to the 3/2.

00:05:52.380 --> 00:05:53.720
Let me make it clear.

00:05:53.720 --> 00:05:59.160
This negative 2 right
here is this negative 2.

00:05:59.160 --> 00:06:02.130
And this expression
right over here

00:06:02.130 --> 00:06:04.367
is the antiderivative
of square root of x.

00:06:04.367 --> 00:06:05.450
And you could verify this.

00:06:05.450 --> 00:06:08.520
If you take the derivative
of this, 3/2 times 2/3 is 1.

00:06:08.520 --> 00:06:11.690
Decrement the 3/2,
you get x to the 1/2.

00:06:11.690 --> 00:06:13.930
Now let's take the
antiderivative of 1.

00:06:13.930 --> 00:06:17.000
Well, that's just
going to be equal to x.

00:06:17.000 --> 00:06:20.422
And we're going to
evaluate this from 1 to 4.

00:06:20.422 --> 00:06:21.630
So we're in the home stretch.

00:06:21.630 --> 00:06:26.720
So this is going to
be equal to pi times--

00:06:26.720 --> 00:06:28.630
well, first let's do it at 4.

00:06:28.630 --> 00:06:31.170
So you have 4 squared--
I'll write it all out.

00:06:31.170 --> 00:06:36.600
4 squared over 2 minus-- what's
4 to the 3-- well, let me

00:06:36.600 --> 00:06:37.650
do this part first.

00:06:37.650 --> 00:06:40.920
So minus 4/3.

00:06:40.920 --> 00:06:44.710
I know it's 4 to the 3/2,
so 4 to the 1/2 is 2,

00:06:44.710 --> 00:06:47.290
and then you raise that to
the third power, you get 8.

00:06:47.290 --> 00:06:52.730
So times 8 plus 4.

00:06:52.730 --> 00:06:55.830
And then you subtract out all
of this stuff evaluated at 1.

00:06:55.830 --> 00:07:00.450
So 1 squared over 2, so that's
just going to be minus 1/2.

00:07:00.450 --> 00:07:02.570
And then we're
subtracting this now.

00:07:02.570 --> 00:07:05.050
Actually, I don't want
to skip too many steps.

00:07:05.050 --> 00:07:08.615
So it's going to be this.

00:07:08.615 --> 00:07:10.299
This is when we
evaluate it at 4.

00:07:10.299 --> 00:07:11.840
And then we're going
to subtract when

00:07:11.840 --> 00:07:16.270
we evaluate this
whole thing at 1.

00:07:16.270 --> 00:07:19.130
So when we evaluate
it at 1-- let

00:07:19.130 --> 00:07:22.540
me do this in this green color.

00:07:22.540 --> 00:07:26.140
That's not the green color
I thought I was going to do.

00:07:26.140 --> 00:07:29.960
So when you evaluate
it at 1, you

00:07:29.960 --> 00:07:32.750
get 1 squared over
2, which is 1/2.

00:07:32.750 --> 00:07:35.040
And then you get 1 to
the 3/2, which is just 1.

00:07:35.040 --> 00:07:36.470
So this becomes minus 4/3.

00:07:39.240 --> 00:07:41.926
And then you have plus 1.

00:07:41.926 --> 00:07:43.050
And so let's simplify this.

00:07:43.050 --> 00:07:45.360
And I'll do it all in
the same color now.

00:07:45.360 --> 00:07:48.870
This is going to be
equal to pi times-- well,

00:07:48.870 --> 00:07:55.460
4 squared over 2, that is
16/2, which is equal to 8.

00:07:55.460 --> 00:08:00.300
And then you have 4 times
8, which is 32, over 3.

00:08:00.300 --> 00:08:04.860
So minus 32/3, plus 4.

00:08:04.860 --> 00:08:06.900
And then you have
minus 1/2-- we're just

00:08:06.900 --> 00:08:13.396
distributing the negative-- plus
4/3, and then you have minus 1.

00:08:13.396 --> 00:08:15.520
And now we just have to
add up a bunch of fractions

00:08:15.520 --> 00:08:16.670
to simplify this thing.

00:08:16.670 --> 00:08:17.990
So what do we get?

00:08:17.990 --> 00:08:21.410
This is going to be equal
to pi times-- let's see,

00:08:21.410 --> 00:08:23.340
our least common
multiple looks like 6.

00:08:23.340 --> 00:08:25.905
We're going to put everything
over a denominator of 6.

00:08:29.300 --> 00:08:33.243
And so 8 is the
same thing as 48/6.

00:08:33.243 --> 00:08:39.740
32/3 is the same thing
as 64/6, so minus 64/6.

00:08:39.740 --> 00:08:43.640
4 is the same thing as 24/6.

00:08:43.640 --> 00:08:47.790
1/2 is the same thing as
3/6, so this is negative 3/6,

00:08:47.790 --> 00:08:50.010
the same thing as negative 1/2.

00:08:50.010 --> 00:08:53.680
4/3 is the same thing as 8/6.

00:08:53.680 --> 00:08:58.040
And then negative 1 is the
same thing as negative 6/6.

00:08:58.040 --> 00:09:03.350
A little bit of arithmetic
here and let's see what we get.

00:09:03.350 --> 00:09:10.790
So if we take 48 and we subtract
64 from 48, we get negative 16.

00:09:10.790 --> 00:09:11.660
Is that right?

00:09:11.660 --> 00:09:14.220
Yes, we get negative 16.

00:09:14.220 --> 00:09:17.100
And then if we add
24 to negative 16,

00:09:17.100 --> 00:09:18.940
you get positive 8.

00:09:18.940 --> 00:09:21.950
Positive 8 minus 3 is 5.

00:09:21.950 --> 00:09:23.350
5 plus 8 is 13.

00:09:23.350 --> 00:09:26.370
13 minus 6 is 7.

00:09:26.370 --> 00:09:28.930
So this whole numerator
simplifies to 7.

00:09:28.930 --> 00:09:34.094
So we get 7 pi over
6 as our volume.

00:09:34.094 --> 00:09:35.760
And let me just verify
I did that right.

00:09:35.760 --> 00:09:38.030
So this is going
to be negative 16.

00:09:38.030 --> 00:09:41.300
We get to positive 8, positive
8 plus positive 8 is 16.

00:09:41.300 --> 00:09:45.820
16 minus 9 is--
yes, it is indeed 7.

00:09:45.820 --> 00:09:48.530
So we get 7 pi over
6, and we're done.

00:09:48.530 --> 00:09:51.280
We figured out our volume
of this little sideways

00:09:51.280 --> 00:09:54.070
cone-looking thing.