Generalizing the washer method

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.130
Let's now generalize what
we did in the last video.

00:00:03.130 --> 00:00:10.080
So if this is my y-axis and this
is-- that's not that straight.

00:00:10.080 --> 00:00:14.037
This right over
here is my x-axis.

00:00:14.037 --> 00:00:15.495
And let's say I
have two functions.

00:00:15.495 --> 00:00:18.160
So I'm just going to
say it in general terms.

00:00:18.160 --> 00:00:20.700
So let's say I have a
function right over here.

00:00:20.700 --> 00:00:23.860
So let's say it looks
something like this.

00:00:23.860 --> 00:00:25.200
So that's one function.

00:00:25.200 --> 00:00:28.850
So this is y is equal to f of x.

00:00:28.850 --> 00:00:32.420
And then I have another function
that y is equal to g of x.

00:00:32.420 --> 00:00:35.090
So let's say it looks
something like this.

00:00:35.090 --> 00:00:41.484
So the blue one right over
here is y is equal to g of x.

00:00:41.484 --> 00:00:42.900
And like we did
in the last video,

00:00:42.900 --> 00:00:46.480
I want to think about the volume
of the solid of revolution

00:00:46.480 --> 00:00:49.380
we will get if we
essentially rotate

00:00:49.380 --> 00:00:53.480
the area between
these two, and if we

00:00:53.480 --> 00:00:56.470
were to rotate it
around the x-axis.

00:00:56.470 --> 00:00:58.224
So we're saying in
very general terms.

00:00:58.224 --> 00:00:59.390
This could be anything, But.

00:00:59.390 --> 00:01:01.390
In the way we've draw it
now, it would literally

00:01:01.390 --> 00:01:02.970
be that same truffle shape.

00:01:02.970 --> 00:01:04.690
It would be a very
similar truffle shape,

00:01:04.690 --> 00:01:06.900
where on the outside it
looks like a truffle.

00:01:06.900 --> 00:01:09.040
On the outside, it looks
something like a truffle,

00:01:09.040 --> 00:01:11.720
and on the inside, we
have carved out a cone.

00:01:11.720 --> 00:01:14.500
Obviously, this visualization
is very specific to the way

00:01:14.500 --> 00:01:16.520
I've drawn these
functions, but what

00:01:16.520 --> 00:01:19.840
we want to do is generalize at
least the mathematics of it.

00:01:19.840 --> 00:01:21.860
So how do we find a volume?

00:01:21.860 --> 00:01:23.606
Well, we could think of disks.

00:01:23.606 --> 00:01:24.980
But instead of
thinking of disks,

00:01:24.980 --> 00:01:26.930
we're going to think
about washers now,

00:01:26.930 --> 00:01:28.780
which is essentially
the exact same thing we

00:01:28.780 --> 00:01:30.321
did in the last
video mathematically,

00:01:30.321 --> 00:01:33.120
but it's a slightly different
way of conceptualizing it.

00:01:33.120 --> 00:01:43.910
So imagine taking a little chunk
between these two functions,

00:01:43.910 --> 00:01:45.070
just like that.

00:01:45.070 --> 00:01:50.750
What is going to be the
width of this chunk?

00:01:50.750 --> 00:01:54.110
Well, it's going
to be equal to dx.

00:01:54.110 --> 00:01:57.120
And let's rotate that whole
thing around the x-axis.

00:01:57.120 --> 00:01:59.260
So if we rotate this
thing around the x-axis,

00:01:59.260 --> 00:02:00.620
we end up with a washer.

00:02:00.620 --> 00:02:03.080
That's why we're going to
call this the washer method.

00:02:03.080 --> 00:02:05.500
And it's really just
kind of the disk method,

00:02:05.500 --> 00:02:08.610
where you're gutting out
the inside of a disk.

00:02:08.610 --> 00:02:10.830
So that's the inside
of our washer.

00:02:10.830 --> 00:02:13.690
And then this is the
outside of our washer.

00:02:13.690 --> 00:02:19.530
Outside of our washer
looks something like that.

00:02:19.530 --> 00:02:22.470
Hopefully, that makes sense.

00:02:22.470 --> 00:02:27.500
And so the surface of our washer
looks something like that.

00:02:27.500 --> 00:02:29.810
I know I could have drawn
this a little bit better,

00:02:29.810 --> 00:02:32.240
but hopefully it
serves the purpose

00:02:32.240 --> 00:02:33.290
so that we understand it.

00:02:33.290 --> 00:02:35.870
So the surface of our washer
looks something like that.

00:02:35.870 --> 00:02:37.160
And it has depth of dx.

00:02:40.410 --> 00:02:43.270
So let me see how
well I can draw this.

00:02:43.270 --> 00:02:44.340
So depth dx.

00:02:44.340 --> 00:02:46.790
That's the side of this washer.

00:02:46.790 --> 00:02:52.730
So a washer, you can imagine,
is kind of a gutted out coin.

00:02:52.730 --> 00:02:54.410
So how do we find the volume?

00:02:54.410 --> 00:02:56.242
Well once again, if
we know the surface,

00:02:56.242 --> 00:02:58.200
if we know the area of
the face of this washer,

00:02:58.200 --> 00:03:00.600
we can just multiply
that times the depth.

00:03:00.600 --> 00:03:03.440
So it's going to be the area
of the face of the washer.

00:03:03.440 --> 00:03:06.020
So the area of the
face of the washer--

00:03:06.020 --> 00:03:09.610
well, it could be the area
if it wasn't gutted out.

00:03:09.610 --> 00:03:11.220
And what would that area be?

00:03:11.220 --> 00:03:14.440
Well, it would be pi
times the overall,

00:03:14.440 --> 00:03:16.360
the outside radius squared.

00:03:16.360 --> 00:03:19.680
It would be pi times the
outside radius squared.

00:03:19.680 --> 00:03:21.040
Well, what's the outside radius?

00:03:21.040 --> 00:03:23.170
The radius that goes to
the outside of the washer?

00:03:23.170 --> 00:03:25.260
Well, that's f of x.

00:03:25.260 --> 00:03:28.190
So it's going to be
f of x is the radius.

00:03:28.190 --> 00:03:30.060
And we're just going
to square that.

00:03:30.060 --> 00:03:32.050
So this expression
right over here

00:03:32.050 --> 00:03:34.340
would give us the area
of the entire face

00:03:34.340 --> 00:03:36.350
if it wasn't a washer,
if it was a coin.

00:03:36.350 --> 00:03:39.150
But now we have to
subtract out the inside.

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So what's the area
of the inside?

00:03:43.410 --> 00:03:45.599
This part right over here?

00:03:45.599 --> 00:03:47.140
Well, we're going
to subtract it out.

00:03:47.140 --> 00:03:51.500
It's going to be pi times the
radius of the inside squared.

00:03:54.460 --> 00:03:56.980
Well, what's the radius
of the inside squared?

00:03:56.980 --> 00:03:59.940
Well, the inside, in
this case, is g of x.

00:03:59.940 --> 00:04:04.590
It's going to be pi
times g of x squared.

00:04:04.590 --> 00:04:06.940
That's the inner function,
at least over the interval

00:04:06.940 --> 00:04:07.980
that we care about.

00:04:07.980 --> 00:04:11.000
So the area of this washer, we
could just leave it like this,

00:04:11.000 --> 00:04:12.970
or we could factor out a pi.

00:04:12.970 --> 00:04:15.676
We could say the area is
equal to-- if we factor out

00:04:15.676 --> 00:04:25.546
a pi-- pi times f of x
squared minus g of x squared.

00:04:25.546 --> 00:04:27.640
I don't really have to
write a parentheses there.

00:04:27.640 --> 00:04:32.880
So we could write f of x
squared minus g of x squared.

00:04:32.880 --> 00:04:36.160
And then if we want the volume--
put that in the same yellow.

00:04:36.160 --> 00:04:37.890
If we want the
volume of this thing,

00:04:37.890 --> 00:04:41.500
we just multiply it times the
depth of each of those washers.

00:04:41.500 --> 00:04:43.680
So the volume of
each of these washers

00:04:43.680 --> 00:04:54.060
are going to be pi times f of
x squared minus g of x squared.

00:04:54.060 --> 00:04:56.410
The outer function
squared over our interval,

00:04:56.410 --> 00:04:59.160
minus our inner function
squared over the interval,

00:04:59.160 --> 00:05:00.320
and then times our depth.

00:05:03.380 --> 00:05:05.990
That'd be the volume of
each of these washers.

00:05:05.990 --> 00:05:09.150
And that's going to be defined
at a given x in our interval,

00:05:09.150 --> 00:05:10.550
but for each x at
these interval,

00:05:10.550 --> 00:05:11.760
we can define a new washer.

00:05:11.760 --> 00:05:14.219
So there could be a washer out
here, and a washer out here.

00:05:14.219 --> 00:05:16.551
And so we're going to take
the sum of all those washers,

00:05:16.551 --> 00:05:18.960
and take the limit as we have
smaller and smaller depths.

00:05:18.960 --> 00:05:22.290
And we have an infinite number
of infinitely thin washers.

00:05:22.290 --> 00:05:27.130
So we're going take the integral
over our interval from where

00:05:27.130 --> 00:05:29.870
these two things intersect, the
interval that we care about.

00:05:29.870 --> 00:05:31.790
It doesn't have to be where they
intersect, but in this case,

00:05:31.790 --> 00:05:32.970
that's what we'll do.

00:05:32.970 --> 00:05:35.480
So let's say x equals
a to x equals b.

00:05:35.480 --> 00:05:38.709
Although this could have been
a, that could have been b.

00:05:38.709 --> 00:05:41.000
But this is our interval
we're saying in general terms,

00:05:41.000 --> 00:05:43.710
from a to b.

00:05:43.710 --> 00:05:45.770
And this will give our volume.

00:05:45.770 --> 00:05:49.070
This right over here is
the volume of each washer.

00:05:49.070 --> 00:05:50.930
And then we're summing
up all of the washers

00:05:50.930 --> 00:05:54.020
and taking the limit, as we
have an infinite number of them.

00:05:54.020 --> 00:05:56.990
So let's see if we apply this
to the example in the last video

00:05:56.990 --> 00:05:59.380
whether we get the
exact same answer.

00:05:59.380 --> 00:06:01.800
Well, in the last
video, y equals

00:06:01.800 --> 00:06:06.340
g of x was equal to
x, and y is equal to f

00:06:06.340 --> 00:06:09.670
of x was equal to
the square root of x.

00:06:09.670 --> 00:06:12.340
So let's evaluate that
given what we just

00:06:12.340 --> 00:06:14.180
were able to derive.

00:06:14.180 --> 00:06:16.070
So our volume-- do it up here.

00:06:16.070 --> 00:06:17.980
The volume is going to
be the integral-- what

00:06:17.980 --> 00:06:19.825
are the two intersection points?

00:06:19.825 --> 00:06:21.900
Well, over here,
once again, we could

00:06:21.900 --> 00:06:23.812
have defined the
interval someplace else,

00:06:23.812 --> 00:06:25.020
like between there and there.

00:06:25.020 --> 00:06:26.520
And we would have gotten
a different shape.

00:06:26.520 --> 00:06:29.000
But the points that we care
about the way we visualize

00:06:29.000 --> 00:06:34.844
it is between x is equal
to 0 and x is equal to 1.

00:06:34.844 --> 00:06:36.510
That's where these
two things intersect.

00:06:36.510 --> 00:06:38.870
We saw that in the last video.

00:06:38.870 --> 00:06:45.290
Of pi times-- what's f of
x squared? f of x squared,

00:06:45.290 --> 00:06:51.220
square root of x squared is
just x, minus g of x squared.

00:06:51.220 --> 00:06:56.560
g of x is x, that
squared is x squared.

00:06:56.560 --> 00:06:59.200
And then we multiply times dx.

00:06:59.200 --> 00:07:02.720
So this is going to be equal
to-- we can factor out the pi.

00:07:02.720 --> 00:07:09.550
0 to 1, x minus x squared dx,
which is equal to pi times--

00:07:09.550 --> 00:07:13.005
let's see, the antiderivative
of x is x squared over 2.

00:07:13.005 --> 00:07:17.600
The antiderivative of x squared
is x squared over 3-- sorry, x

00:07:17.600 --> 00:07:19.770
to the third over 3.

00:07:19.770 --> 00:07:23.290
And we're going to
evaluate this from 0 to 1.

00:07:23.290 --> 00:07:25.630
So this is going to be
equal to-- I'm running out

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of my real estate a little bit.

00:07:27.180 --> 00:07:30.400
Let me scroll over to
the right a little bit.

00:07:30.400 --> 00:07:34.830
So this is going to be
equal to pi times-- well,

00:07:34.830 --> 00:07:36.880
when you evaluate
this whole thing at 1,

00:07:36.880 --> 00:07:39.510
you get-- let's see,
you get 1/2 minus 1/3.

00:07:42.730 --> 00:07:44.700
And then you subtract
it, evaluate it at 0,

00:07:44.700 --> 00:07:45.950
but that's just going to be 0.

00:07:45.950 --> 00:07:50.490
0 squared over 2 minus
0, 0 to the third power.

00:07:50.490 --> 00:07:52.000
That's just all going to be 0.

00:07:52.000 --> 00:07:53.890
So when you subtract
out 0, you're

00:07:53.890 --> 00:07:56.200
just left with this
expression right over here.

00:07:56.200 --> 00:07:58.000
What's 1/2 minus 1/3?

00:07:58.000 --> 00:07:59.580
Well, that's 1/6.

00:07:59.580 --> 00:08:01.890
This is 1/6.

00:08:01.890 --> 00:08:05.310
And so we're left with
this is equal to pi over 6,

00:08:05.310 --> 00:08:07.810
which is the exact same thing
that we got in the last video.

00:08:07.810 --> 00:08:09.360
And that's because we
did the exact same thing

00:08:09.360 --> 00:08:10.670
that we did in the last video.

00:08:10.670 --> 00:08:13.080
We just conceptualized it
a little bit differently.

00:08:13.080 --> 00:08:15.830
We generalized it in terms
of f of x and g of x.

00:08:15.830 --> 00:08:22.160
And we essentially
conceptualized it as a washer,

00:08:22.160 --> 00:08:26.530
as opposed to doing the disk
method for an outer shape

00:08:26.530 --> 00:08:29.873
and an inner shape like
we did in the last video.


Solid of revolution between two functions (leading up to the washer method)

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:05.220
Let's do some more volumes
of solids of revolutions.

00:00:05.220 --> 00:00:07.460
So let's say that
I have the graph

00:00:07.460 --> 00:00:09.210
y is equal to square root of x.

00:00:09.210 --> 00:00:13.520
So let's do it, so it
looks something like this.

00:00:13.520 --> 00:00:20.060
So that right over there is y is
equal to the square root of x.

00:00:20.060 --> 00:00:24.410
And let's say I also have
the graph of y equals x.

00:00:24.410 --> 00:00:28.220
So let's say y equals x
looks something like this.

00:00:28.220 --> 00:00:30.030
It looks just like that.

00:00:30.030 --> 00:00:32.640
y equals x.

00:00:32.640 --> 00:00:36.700
And what I care about
now is the solid

00:00:36.700 --> 00:00:42.570
I get if I were to rotate the
area between these two things

00:00:42.570 --> 00:00:44.932
around to the x-axis.

00:00:44.932 --> 00:00:46.140
So let's try to visualize it.

00:00:46.140 --> 00:00:50.244
So the outside is going to
be kind of a truffle shape,

00:00:50.244 --> 00:00:51.910
it's going to look
like a truffle shape.

00:00:51.910 --> 00:00:55.220
And then we hollow out
a cone inside of it.

00:00:55.220 --> 00:00:58.160
So I'll make my best
attempt to draw this shape.

00:00:58.160 --> 00:01:05.170
So it's going to look something
like-- so the outside is

00:01:05.170 --> 00:01:08.180
going to look
something like this.

00:01:08.180 --> 00:01:12.230
It's going to look
something like that.

00:01:12.230 --> 00:01:13.800
And we care about the interval.

00:01:13.800 --> 00:01:16.080
We care about the interval
between the points

00:01:16.080 --> 00:01:17.030
that they intersect.

00:01:17.030 --> 00:01:19.250
So between this point
and this point here.

00:01:19.250 --> 00:01:22.125
So the outside is going to
look something like this.

00:01:22.125 --> 00:01:24.749
So this is the base
of the truffle.

00:01:24.749 --> 00:01:26.290
That is the base of
the truffle, it's

00:01:26.290 --> 00:01:29.330
going to have this kind of
truffle shape on the outside.

00:01:29.330 --> 00:01:31.350
But I guess maybe we're
on some type of a diet,

00:01:31.350 --> 00:01:33.510
we don't want to eat
to the entire truffle.

00:01:33.510 --> 00:01:36.500
So we carve out a
cone on the inside.

00:01:36.500 --> 00:01:38.800
So the inside of
it is essentially

00:01:38.800 --> 00:01:40.800
hollow except for this
kind of shell part.

00:01:40.800 --> 00:01:48.090
So we carve out a
cone in the center.

00:01:48.090 --> 00:01:50.650
So we rotate it
around to the x-axis.

00:01:50.650 --> 00:01:54.422
Truffle on the outside, carved
out a cone on the inside.

00:01:54.422 --> 00:01:56.380
So what's going to be
the volume of that thing?

00:01:56.380 --> 00:02:01.350
So it's essentially, if we
take a slice of our figure,

00:02:01.350 --> 00:02:04.312
it's going to be, this
is going to be the wall.

00:02:04.312 --> 00:02:05.770
And we're essentially
going to take

00:02:05.770 --> 00:02:08.440
the volume of this
entire wall that we're

00:02:08.440 --> 00:02:11.520
rotating around the x-axis.

00:02:11.520 --> 00:02:14.580
So how do we do that?

00:02:14.580 --> 00:02:16.360
Well, it might dawn
on you that if we

00:02:16.360 --> 00:02:19.210
found the volume of the
truffle if it was not

00:02:19.210 --> 00:02:22.560
carved out, and then
subtract from that the volume

00:02:22.560 --> 00:02:24.600
of the cone, we
would essentially

00:02:24.600 --> 00:02:29.350
find out the volume of the
space in between the outside

00:02:29.350 --> 00:02:32.560
of the truffle and the
cone part of the truffle.

00:02:32.560 --> 00:02:34.370
So how would we do that?

00:02:34.370 --> 00:02:36.860
Well, so to find the
volume of the outer shape.

00:02:36.860 --> 00:02:38.441
So let me draw it over here.

00:02:38.441 --> 00:02:40.190
So actually let me
just draw it over here.

00:02:40.190 --> 00:02:42.189
If we think about the
volume of the outer shape,

00:02:42.189 --> 00:02:44.180
once again, we can
use the disk method.

00:02:44.180 --> 00:02:48.840
So at any given point in time
our radius for one of our disks

00:02:48.840 --> 00:02:50.950
is going to be equal
to the function.

00:02:50.950 --> 00:02:53.659
Let's rotate that disk around.

00:02:53.659 --> 00:02:55.450
Actually let me do it
in a different color,

00:02:55.450 --> 00:02:57.045
it's hard to see that
disk because it's

00:02:57.045 --> 00:02:57.878
in the same magenta.

00:02:57.878 --> 00:03:01.890
So this is a radius, and
let's rotate it around.

00:03:01.890 --> 00:03:04.860
So I'm rotating the disk around.

00:03:04.860 --> 00:03:08.280
This is our face of the disk,
that's our face of the disk.

00:03:08.280 --> 00:03:10.820
It's going to have
a depth of dx.

00:03:10.820 --> 00:03:12.520
We've seen this multiple times.

00:03:12.520 --> 00:03:15.990
It's going to have
a depth of dx.

00:03:15.990 --> 00:03:18.980
So the volume of
this disk is going

00:03:18.980 --> 00:03:23.090
to be our depth, dx, times
the area of the face.

00:03:23.090 --> 00:03:26.730
The area of the face is going to
be pi times the radius squared.

00:03:26.730 --> 00:03:30.430
The radius is going to
be equal to the value

00:03:30.430 --> 00:03:32.820
of the outer function, in this
case the square root of x.

00:03:32.820 --> 00:03:35.760
So it's going to be pi
times our radius squared,

00:03:35.760 --> 00:03:40.150
which is pi times the
square root of x squared.

00:03:40.150 --> 00:03:41.950
And so if we want
to find the volume

00:03:41.950 --> 00:03:45.469
of the entire outer
thimble, or truffle,

00:03:45.469 --> 00:03:46.760
or whatever we want to call it.

00:03:46.760 --> 00:03:50.130
Before we even carve
out the center,

00:03:50.130 --> 00:03:54.409
we just take a sum of a bunch
of these, a bunch of these disks

00:03:54.409 --> 00:03:55.200
that we've created.

00:03:55.200 --> 00:03:58.592
So that's one disk, we would
have another disk over here.

00:03:58.592 --> 00:03:59.550
Another disk over here.

00:03:59.550 --> 00:04:01.510
For each x we have another disk.

00:04:01.510 --> 00:04:03.320
And as we go, as x's
get larger and larger,

00:04:03.320 --> 00:04:06.030
the disks have a larger
and larger radius.

00:04:06.030 --> 00:04:09.700
So we're going to sum
up all of those disks.

00:04:09.700 --> 00:04:12.040
And we're going to take the
limit as each of those disks

00:04:12.040 --> 00:04:15.037
get infinitely thin, and we
have an infinite number of them.

00:04:15.037 --> 00:04:17.370
But we have to figure out our
boundaries of integration.

00:04:17.370 --> 00:04:19.119
So what are our
boundaries of integration?

00:04:19.119 --> 00:04:22.865
What are the two points right
over here where they intersect?

00:04:22.865 --> 00:04:24.490
Well we could just
set these two things

00:04:24.490 --> 00:04:25.940
to be equal to each other.

00:04:25.940 --> 00:04:29.140
If you just said x is
equal to square root of x,

00:04:29.140 --> 00:04:32.482
when does x equal
the square root of x?

00:04:32.482 --> 00:04:34.190
I mean you can square
both sides of this.

00:04:34.190 --> 00:04:37.980
You could say when
does x squared equal x?

00:04:37.980 --> 00:04:40.590
You could, well we
could keep it there.

00:04:40.590 --> 00:04:42.260
You could kind of
solve this, there's

00:04:42.260 --> 00:04:43.510
multiple ways you could do it.

00:04:43.510 --> 00:04:45.950
But you could solve this kind
of just thinking about it.

00:04:45.950 --> 00:04:48.030
If x is equal to 0, x
squared is equal to x.

00:04:48.030 --> 00:04:50.030
And you see that on the
graph right over here. x

00:04:50.030 --> 00:04:51.210
is equal to 0.

00:04:51.210 --> 00:04:53.370
And also 1 squared
is equal to 1.

00:04:53.370 --> 00:04:55.666
1 is equal to the
square root of 1.

00:04:55.666 --> 00:04:57.040
You could have
done other things.

00:04:57.040 --> 00:05:00.580
You could say OK, x squared
minus x is equal to 0.

00:05:00.580 --> 00:05:01.670
You could factor out an x.

00:05:01.670 --> 00:05:05.330
You get x times x
minus 1 is equal to 0.

00:05:05.330 --> 00:05:08.000
And so either one of
these could be equal to 0,

00:05:08.000 --> 00:05:12.110
so x is equal to 0 or x
minus 1 is equal to 0.

00:05:12.110 --> 00:05:16.290
And then you get x equals
0 or x is equal to 1.

00:05:19.520 --> 00:05:21.920
Which gives us our
boundaries of integration.

00:05:21.920 --> 00:05:25.340
So this goes from x
equals 0 to x equals 1.

00:05:25.340 --> 00:05:27.740
And so for the
outside of our shape

00:05:27.740 --> 00:05:30.250
we can now figure
out the volume.

00:05:30.250 --> 00:05:31.120
But we're not done.

00:05:31.120 --> 00:05:32.661
We also need to
figure out the volume

00:05:32.661 --> 00:05:35.040
of the inside of our shape
that we're going to take out.

00:05:35.040 --> 00:05:37.212
So we're going to
subtract out that volume.

00:05:37.212 --> 00:05:38.920
So we're going to
subtract out of volume.

00:05:38.920 --> 00:05:42.400
Our x values, once again,
are going between 0 and 1.

00:05:42.400 --> 00:05:44.930
And so let's think
about those disks.

00:05:44.930 --> 00:05:46.820
So let's think about,
let's construct a disk

00:05:46.820 --> 00:05:49.500
on the inside, right over here.

00:05:49.500 --> 00:05:52.580
So if I construct a
disk on the inside.

00:05:52.580 --> 00:05:55.660
So now I'm carving out
the cone part of it.

00:05:55.660 --> 00:06:00.660
What is the area of the
face of one of those disks?

00:06:00.660 --> 00:06:06.070
Well it's going to be pi
times the radius squared.

00:06:06.070 --> 00:06:08.050
In this case, the
radius is going

00:06:08.050 --> 00:06:10.300
to be equal to the value of
this inner function, which

00:06:10.300 --> 00:06:13.070
is just x.

00:06:13.070 --> 00:06:15.150
And so this is just
y is equal to x.

00:06:15.150 --> 00:06:18.130
And then we're going to
multiply it times the depth,

00:06:18.130 --> 00:06:20.730
times the depth of
each of these disks.

00:06:20.730 --> 00:06:24.345
And each of these disks are
going to have a depth of dx.

00:06:24.345 --> 00:06:29.610
If you imagine a quarter that
has an infinitely-thin depth

00:06:29.610 --> 00:06:30.450
right over here.

00:06:30.450 --> 00:06:32.360
So it's going to be dx.

00:06:32.360 --> 00:06:37.200
And so the volume our, kind of
our truffle with a cone carved

00:06:37.200 --> 00:06:38.730
out, is going to
be this integral

00:06:38.730 --> 00:06:40.740
minus this integral
right over here.

00:06:40.740 --> 00:06:42.860
And we could evaluate
it just like that.

00:06:42.860 --> 00:06:44.980
Or we could even say,
OK we could factor out

00:06:44.980 --> 00:06:46.542
a pi out of both of them.

00:06:46.542 --> 00:06:48.250
There are actually,
there's multiple ways

00:06:48.250 --> 00:06:48.630
that we could write it.

00:06:48.630 --> 00:06:50.171
But let's just
evaluate it like this,

00:06:50.171 --> 00:06:52.380
and then I'll generalize
it in the next video.

00:06:52.380 --> 00:06:55.770
So this is going to be equal
to the definite integral from 0

00:06:55.770 --> 00:06:56.610
to 1.

00:06:56.610 --> 00:06:58.340
You take the pi outside.

00:06:58.340 --> 00:06:59.980
Square root of x
squared is going

00:06:59.980 --> 00:07:06.650
to be x dx minus the integral,
we can factor the pi out.

00:07:06.650 --> 00:07:12.410
From 0 to 1 of x squared dx.

00:07:12.410 --> 00:07:15.190
And we could say
this is going to be

00:07:15.190 --> 00:07:21.620
equal to pi times the
antiderivative of x, which

00:07:21.620 --> 00:07:26.290
is just x squared over
2 evaluated from 0 to 1,

00:07:26.290 --> 00:07:30.390
minus pi, times the
antiderivative of x squared,

00:07:30.390 --> 00:07:35.900
which is x to the third over
3 evaluated from 0 to 1.

00:07:35.900 --> 00:07:37.500
This expression is
equal to-- and I'm

00:07:37.500 --> 00:07:39.050
going to arbitrarily
switch colors

00:07:39.050 --> 00:07:43.410
just because the green's getting
monotonous-- pi times 1 squared

00:07:43.410 --> 00:07:47.050
over 2 minus 0 squared over 2.

00:07:47.050 --> 00:07:48.400
I could write squared.

00:07:48.400 --> 00:07:55.530
1 squared over 2 minus 0
squared over 2, minus pi times 1

00:07:55.530 --> 00:08:01.000
to the third over 3 minus
0 to the third over 3.

00:08:01.000 --> 00:08:04.970
And so we get, this
is equal to-- let

00:08:04.970 --> 00:08:08.890
me do it in that
same blue color--

00:08:08.890 --> 00:08:10.330
so this is this simplified.

00:08:10.330 --> 00:08:11.800
This is just 0 right over here.

00:08:11.800 --> 00:08:15.050
This is 1 squared over
2, which is just 1/2.

00:08:15.050 --> 00:08:21.360
So it's just pi over 2, 1/2
times pi minus-- well this

00:08:21.360 --> 00:08:26.560
is just 0, this is
1/3, minus pi over 3.

00:08:26.560 --> 00:08:28.080
And then to simplify
this, it's just

00:08:28.080 --> 00:08:30.290
really subtracting fractions.

00:08:30.290 --> 00:08:32.700
So we can find a
common denominator.

00:08:32.700 --> 00:08:34.970
Common denominator is 6.

00:08:34.970 --> 00:08:38.570
This is going to be 3 pi over 6.

00:08:38.570 --> 00:08:42.544
This is 3 pi over 6
minus 2 pi over 6.

00:08:45.610 --> 00:08:50.120
pi over 3 is 2 pi over 6,
pi over 2 is 3 pi over 6.

00:08:50.120 --> 00:08:54.020
And we end up with, we
end up with 3 of something

00:08:54.020 --> 00:08:56.240
minus 2 of something, you
end up with 1 of something.

00:08:56.240 --> 00:08:59.120
We end up with 1 pi over 6.

00:08:59.120 --> 00:09:00.390
And we are done.

00:09:00.390 --> 00:09:03.610
We were able to find the
volume of that wacky kind

00:09:03.610 --> 00:09:06.189
of gutted-out truffle.


Disc method around y-axis

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.100
Here we have the graph,
or part of the graph, of y

00:00:03.100 --> 00:00:04.850
is equal to x squared again.

00:00:04.850 --> 00:00:09.310
And I want to find the volume
of another solid of revolution.

00:00:09.310 --> 00:00:11.910
But instead of rotating
around the x-axis this time,

00:00:11.910 --> 00:00:13.665
I want to rotate
around the y-axis.

00:00:13.665 --> 00:00:15.870
And instead of going
between 0 and some point,

00:00:15.870 --> 00:00:23.320
I'm going to go between y is
equal to 1 and y is equal to 4.

00:00:23.320 --> 00:00:24.820
So what I'm going
to do is I'm going

00:00:24.820 --> 00:00:27.720
to take this graph
right over here.

00:00:27.720 --> 00:00:29.310
I'm going to take this curve.

00:00:29.310 --> 00:00:31.950
And instead of rotating it
around the x-axis, like we

00:00:31.950 --> 00:00:33.850
did in the last few
videos, I'm going

00:00:33.850 --> 00:00:35.610
to rotate it around the y-axis.

00:00:35.610 --> 00:00:40.960
So I'm going to rotate
it around just like that.

00:00:40.960 --> 00:00:42.750
So what's the shape
that we would get?

00:00:42.750 --> 00:00:44.930
So let me see if we
can visualize this.

00:00:44.930 --> 00:00:46.580
So the base is going
to look something

00:00:46.580 --> 00:00:53.180
like that if we
could see through it.

00:00:53.180 --> 00:00:55.630
And then this up
here, the top of it,

00:00:55.630 --> 00:01:02.270
would look something like that.

00:01:02.270 --> 00:01:04.690
And we care about
the stuff in between.

00:01:04.690 --> 00:01:07.370
So we care about this
part right over here,

00:01:07.370 --> 00:01:09.250
not the very bottom of it.

00:01:09.250 --> 00:01:10.750
And let me shade
it in a little bit.

00:01:10.750 --> 00:01:16.270
So it would look
something like that.

00:01:16.270 --> 00:01:19.280
So let me draw it separately,
just so we can visualize it.

00:01:19.280 --> 00:01:20.930
So I'll draw it at
different angles.

00:01:20.930 --> 00:01:23.780
So if I were to draw it with
the y-axis kind of coming out

00:01:23.780 --> 00:01:26.365
the back, it would look
something like this.

00:01:26.365 --> 00:01:29.110
It would look something like--
it gets a little bit smaller

00:01:29.110 --> 00:01:30.330
like that.

00:01:30.330 --> 00:01:33.910
And then it gets cut off right
over here, right over here

00:01:33.910 --> 00:01:34.480
like that.

00:01:34.480 --> 00:01:37.290
So it looks-- I don't know
what shape you could call it.

00:01:37.290 --> 00:01:39.570
But I think hopefully
you're conceptualizing this.

00:01:39.570 --> 00:01:41.550
Let me do it in that
same yellow color.

00:01:41.550 --> 00:01:43.240
The visual-- that's not yellow.

00:01:43.240 --> 00:01:46.350
The visualization here is
probably the hardest part.

00:01:46.350 --> 00:01:48.100
But as we can see
it's not too bad.

00:01:48.100 --> 00:01:52.490
So it looks something like this.

00:01:52.490 --> 00:01:55.410
It looks like maybe a truffle,
an upside-down truffle.

00:01:55.410 --> 00:01:57.990
So this right here, let
me draw the y-axis just

00:01:57.990 --> 00:01:59.600
to show how we're oriented.

00:01:59.600 --> 00:02:04.200
So the y-axis is popping out
in this example like that.

00:02:04.200 --> 00:02:05.570
Then it goes down over here.

00:02:05.570 --> 00:02:10.070
And then the x-axis
is going like this.

00:02:10.070 --> 00:02:12.140
So I just tilted this over.

00:02:12.140 --> 00:02:13.630
I tilted it over
a little bit to be

00:02:13.630 --> 00:02:15.171
able to view it at
a different angle.

00:02:15.171 --> 00:02:22.111
This top right over here is
this top right over there.

00:02:22.111 --> 00:02:24.110
So that gives you an idea
of what it looks like.

00:02:24.110 --> 00:02:26.318
But we still haven't thought
about how do we actually

00:02:26.318 --> 00:02:28.880
find the volume of this thing?

00:02:28.880 --> 00:02:31.590
Well, what we can do,
instead of creating

00:02:31.590 --> 00:02:34.600
discs where the depth is in
little dx's, what if we created

00:02:34.600 --> 00:02:36.590
discs where the depth is in dy?

00:02:36.590 --> 00:02:38.690
So let's think about
that a little bit.

00:02:38.690 --> 00:02:42.280
So let's create-- let's
think about constructing

00:02:42.280 --> 00:02:45.110
a disc at a certain y-value.

00:02:45.110 --> 00:02:47.230
So let's think about
a certain y-value,

00:02:47.230 --> 00:02:50.360
and we're going to construct
a disc right over there that

00:02:50.360 --> 00:02:54.480
has the same radius of
the shape at that point.

00:02:54.480 --> 00:02:56.800
So that's our disc.

00:02:56.800 --> 00:02:58.720
That's our disc right over here.

00:03:02.610 --> 00:03:05.190
And then it has
a depth-- instead

00:03:05.190 --> 00:03:09.910
of saying it has a depth of dx,
let's say it has a depth of dy.

00:03:09.910 --> 00:03:14.210
So this depth right
over here is dy.

00:03:14.210 --> 00:03:17.539
So what is the volume of
this disc in terms of y?

00:03:17.539 --> 00:03:19.080
And as you could
imagine, we're going

00:03:19.080 --> 00:03:21.690
to do this definite integral,
and it is a definite integral,

00:03:21.690 --> 00:03:23.530
with respect to y.

00:03:23.530 --> 00:03:26.120
So what's the volume
of this thing?

00:03:26.120 --> 00:03:27.620
Well, like we did
in the last video,

00:03:27.620 --> 00:03:30.410
we have to figure out the
area of the top of each

00:03:30.410 --> 00:03:31.220
of these discs.

00:03:31.220 --> 00:03:34.250
Or I guess you could say
the face of this coin.

00:03:34.250 --> 00:03:36.869
Well, to find that
area it's pi r squared.

00:03:36.869 --> 00:03:38.910
If we can figure out this
radius right over here,

00:03:38.910 --> 00:03:40.600
we know the area.

00:03:40.600 --> 00:03:42.290
So what's that radius?

00:03:42.290 --> 00:03:44.430
So to think about that
radius in terms of y,

00:03:44.430 --> 00:03:48.260
we just have to solve
this equation explicitly

00:03:48.260 --> 00:03:49.100
in terms of y.

00:03:49.100 --> 00:03:51.850
So instead of saying it's
y is equal to x squared,

00:03:51.850 --> 00:03:54.220
we can take the principal
root of both sides,

00:03:54.220 --> 00:03:58.010
and we could say that the
square root of y is equal to x.

00:03:58.010 --> 00:04:01.960
And this right over here is only
defined for non-negative y's,

00:04:01.960 --> 00:04:05.600
but that's OK, because we are
in the positive x-axis right

00:04:05.600 --> 00:04:06.480
over here.

00:04:06.480 --> 00:04:11.010
So we can also call of this
function right over here

00:04:11.010 --> 00:04:13.929
x is equal to the
square root of y.

00:04:13.929 --> 00:04:15.970
And we're essentially
looking at this side of it.

00:04:15.970 --> 00:04:20.335
We're not looking at this
stuff right over here.

00:04:20.335 --> 00:04:22.460
So we're only looking at
this side right over here.

00:04:22.460 --> 00:04:25.800
We've now expressed
this graph, this curve,

00:04:25.800 --> 00:04:28.070
as x as a function of y.

00:04:28.070 --> 00:04:31.090
So if we do it that way, what's
our radius right over here?

00:04:31.090 --> 00:04:33.690
Well, our radius right over
here is going to be f of y.

00:04:33.690 --> 00:04:38.090
It's going to be the
square root of y.

00:04:38.090 --> 00:04:41.020
It's going to be the square
root of y is our radius.

00:04:41.020 --> 00:04:42.520
So it's going to
be a function of y.

00:04:42.520 --> 00:04:44.700
I don't want to confuse you if
you thought this was f of x,

00:04:44.700 --> 00:04:45.866
and actually this is f of y.

00:04:45.866 --> 00:04:48.060
No, it would be a function of y.

00:04:48.060 --> 00:04:49.360
We could call it g of y.

00:04:49.360 --> 00:04:52.420
It's going to be the
square root of y.

00:04:52.420 --> 00:04:57.840
So area is equal to
pi r squared, which

00:04:57.840 --> 00:05:00.380
means that the
area of this thing

00:05:00.380 --> 00:05:04.400
is going to be pi times
our radius, radius squared.

00:05:04.400 --> 00:05:09.290
Our radius is square root of y.

00:05:09.290 --> 00:05:15.490
So this thing is going
to be equal to pi--

00:05:15.490 --> 00:05:19.472
the square root of y
squared is just pi times y.

00:05:19.472 --> 00:05:20.930
Now, if we want
the volume, we just

00:05:20.930 --> 00:05:23.870
have to multiply the area of
this surface times the depth,

00:05:23.870 --> 00:05:25.490
times dy.

00:05:25.490 --> 00:05:27.820
So the volume of
each of those discs

00:05:27.820 --> 00:05:30.430
is going to be pi y times dy.

00:05:34.860 --> 00:05:36.580
This gives you the
volume of a disc.

00:05:41.990 --> 00:05:44.130
Now, if we want the volume
of this entire thing,

00:05:44.130 --> 00:05:45.930
we just have to sum
all of these discs

00:05:45.930 --> 00:05:48.650
for all of the y-values
between y is equal to 1

00:05:48.650 --> 00:05:50.440
and y is equal to 4.

00:05:50.440 --> 00:05:51.380
So let's do that.

00:05:51.380 --> 00:05:54.250
So we just take the definite
integral from y is equal to 1

00:05:54.250 --> 00:05:54.939
and y equals 4.

00:05:54.939 --> 00:05:56.480
Just as a reminder,
definite integral

00:05:56.480 --> 00:05:58.540
is a very special type of sum.

00:05:58.540 --> 00:06:00.320
We're summing up
all of these things.

00:06:00.320 --> 00:06:02.190
But we're taking the
limit of that sum

00:06:02.190 --> 00:06:05.400
as these dy's get
shorter or get, I guess,

00:06:05.400 --> 00:06:07.580
squatter and squatter
or smaller and smaller,

00:06:07.580 --> 00:06:11.650
and we have a larger and
larger number of these discs.

00:06:11.650 --> 00:06:13.890
Really, as these
become infinitely small

00:06:13.890 --> 00:06:16.040
and we have an infinite
number of discs,

00:06:16.040 --> 00:06:19.250
so that our sum doesn't
just approximate the volume,

00:06:19.250 --> 00:06:22.190
it actually is the
volume at the limit.

00:06:22.190 --> 00:06:25.430
So to figure out the volume
of this entire thing,

00:06:25.430 --> 00:06:27.450
we just have to evaluate
this definite integral

00:06:27.450 --> 00:06:28.390
in terms of y.

00:06:28.390 --> 00:06:29.949
And so how do we do that?

00:06:29.949 --> 00:06:31.240
What's it going to be equal to?

00:06:31.240 --> 00:06:33.290
Well, we could take
the pi outside.

00:06:33.290 --> 00:06:38.210
It's going to be pi times the
antiderivative of y, which

00:06:38.210 --> 00:06:42.270
is just y squared over 2,
y squared over 2 evaluated

00:06:42.270 --> 00:06:48.540
from 1 to 4, which is
equal to pi times--

00:06:48.540 --> 00:06:50.972
well, if you evaluate it
at 4, you get 16 over 2.

00:06:50.972 --> 00:06:52.430
Let me just write
it out like this.

00:06:52.430 --> 00:06:57.210
4 squared over 2 minus
1 squared over 2,

00:06:57.210 --> 00:07:07.600
which is equal to pi times
16 over 2 is 8, minus 1/2.

00:07:07.600 --> 00:07:10.600
And so we could view
this as 16/2 minus 1/2,

00:07:10.600 --> 00:07:12.570
which is equal to 15/2.

00:07:12.570 --> 00:07:18.420
So this is equal
to 15/2 times pi.

00:07:18.420 --> 00:07:21.626
Or another way of thinking
of it is 7 and 1/2 times pi.

00:07:21.626 --> 00:07:23.000
But this is a
little bit clearer.

00:07:23.000 --> 00:07:23.690
So we're done.

00:07:23.690 --> 00:07:26.570
We found our volume not
rotating around the x-axis,

00:07:26.570 --> 00:07:32.010
but rotating around the y-axis,
which is kind of exciting.


Generalizing disc method around x-axis

WEBVTT
Kind: captions
Language: en

00:00:00.278 --> 00:00:02.444
What we're going to do in this video

00:00:02.444 --> 00:00:04.292
is generalize what we did in the last video.

00:00:04.292 --> 00:00:06.074
And, essentially, end up with the formula

00:00:06.074 --> 00:00:08.213
for rotating something around the X-axis like this using,

00:00:08.213 --> 00:00:11.565
what we call, the "disk method".

00:00:11.565 --> 00:00:16.165
And the point is to show you where that formula inside a calculus textbook actually comes from.

00:00:16.165 --> 00:00:18.879
But it just comes from the same exact principles we did in the last video!

00:00:18.879 --> 00:00:26.086
It's not advised to memorize the formula; I highly recommend against that because you really need to know what's going on.

00:00:26.086 --> 00:00:29.082
It's really better to do it in first principles where you find

00:00:29.082 --> 00:00:31.613
the volume of each of these disks.

00:00:31.613 --> 00:00:34.901
But let's just generalize what we saw in the last video.

00:00:34.901 --> 00:00:37.612
So instead of saying that "y = x^2",

00:00:37.612 --> 00:00:43.281
let's just say that this is the graph, the function that's right over here.

00:00:43.281 --> 00:00:46.715
Let's just generalize it and call it "y=f(x)".

00:00:46.715 --> 00:00:48.814
And instead of saying x is going from 0 to 2

00:00:48.814 --> 00:00:57.445
let's say that we're going between a and b, so these are just two endpoints along the X-axis.

00:00:57.445 --> 00:00:59.762
So how would we find the volume of this?

00:00:59.762 --> 00:01:04.629
Well, just like the last video, we still take a disk just like this.

00:01:04.629 --> 00:01:06.662
But what is the height of the disk?

00:01:06.662 --> 00:01:09.626
The height of the disk is not just x^2 since we've generalized it.

00:01:09.626 --> 00:01:12.597
So the height is simply going to be whatever the height of the function is at that point.

00:01:12.597 --> 00:01:16.808
So the height of the disk is going to be f(x)!

00:01:16.808 --> 00:01:24.532
The area of the space of this disk is going to be πR^2.

00:01:24.532 --> 00:01:29.440
So our radius is f(x), and we're just going to square it.

00:01:29.440 --> 00:01:32.591
That's the area, that's the area of this face right over here.

00:01:32.591 --> 00:01:34.391
What is the volume of the disk?

00:01:34.391 --> 00:01:39.875
We're just going to multiply that by our depth, which going to be dx.

00:01:39.875 --> 00:01:43.795
And we want to take the sum of all of these disks from a to b,

00:01:43.795 --> 00:01:50.127
and we're going to take the sum of them, and we're going to take the limit as the "dx"s get smaller and smaller.

00:01:50.127 --> 00:01:51.911
And we have an infinite number of these disks.

00:01:51.911 --> 00:01:58.290
Thus, we are going to the integral of this from a to b!

00:01:58.290 --> 00:02:06.367
And this right here is the formula that you will see often in a calculus textbook for using the disk method as you're rotating around the X-axis.

00:02:06.367 --> 00:02:12.834
So I just wanted to show you that it comes out of the common sense of finding the volume of this disk.

00:02:12.834 --> 00:02:18.002
The f(x) right over here is just the radius of the disk, so this part over here is really just πR^2;

00:02:18.002 --> 00:02:24.510
we multiply it [area of the circle] times the depth; then we take the sum from a to b of all of the disks.

00:02:24.510 --> 00:02:28.510
And essentially since this is integral is the limit of all the disks getting narrower and narrower. And thus, we have an infinite number of these disks.


Disc method around x-axis

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.230
Over here I've drawn
part of the graph of y

00:00:03.230 --> 00:00:05.550
is equal to x squared.

00:00:05.550 --> 00:00:07.780
And what we're going
to do is use our powers

00:00:07.780 --> 00:00:12.510
of definite integrals to find
volumes instead of just areas.

00:00:12.510 --> 00:00:14.400
So let's review what
we're doing when

00:00:14.400 --> 00:00:16.980
we take just a regular
definite integral.

00:00:16.980 --> 00:00:19.410
So if we take the definite
integral between, say,

00:00:19.410 --> 00:00:24.580
0 and 2 of x squared dx,
what does that represent?

00:00:24.580 --> 00:00:26.040
Well, let's look
at our endpoints.

00:00:26.040 --> 00:00:28.050
So this is x is equal to 0.

00:00:28.050 --> 00:00:33.180
Let's say that this right
over here is x is equal to 2.

00:00:33.180 --> 00:00:36.390
What we're doing is
for each x, we're

00:00:36.390 --> 00:00:39.600
finding a little dx around
it-- so this right over here

00:00:39.600 --> 00:00:41.430
is a little dx.

00:00:41.430 --> 00:00:44.220
And we're multiplying that
dx times our function,

00:00:44.220 --> 00:00:45.970
times x squared.

00:00:45.970 --> 00:00:48.910
So what we're doing is we're
multiplying this width times

00:00:48.910 --> 00:00:52.440
this height right over here.

00:00:52.440 --> 00:00:55.110
The height right over
here is x squared.

00:00:55.110 --> 00:00:58.180
And we're getting the area
of this little rectangle.

00:00:58.180 --> 00:01:00.740
And the integral
sign is literally

00:01:00.740 --> 00:01:05.940
the sum of all of these
rectangles for all of the x's

00:01:05.940 --> 00:01:09.290
between x is equal to
0 and x is equal to 2.

00:01:09.290 --> 00:01:13.530
But the limit of that as these
dx's get smaller and smaller

00:01:13.530 --> 00:01:16.160
and smaller, get
infinitely small,

00:01:16.160 --> 00:01:17.990
but not being equal to 0.

00:01:17.990 --> 00:01:19.870
And we have an infinite
number of them.

00:01:19.870 --> 00:01:22.490
That's the whole power
of the definite integral.

00:01:22.490 --> 00:01:25.190
And so you can imagine, as these
dx's get smaller and smaller

00:01:25.190 --> 00:01:27.550
and smaller, these rectangles
get narrower and narrower

00:01:27.550 --> 00:01:29.174
and narrower, and we
have more of them,

00:01:29.174 --> 00:01:31.500
we are getting a better
and better approximation

00:01:31.500 --> 00:01:33.780
of the area under the
curve until, at the limit,

00:01:33.780 --> 00:01:36.590
we are getting the
area under the curve.

00:01:36.590 --> 00:01:38.780
Now we're going to apply
that same idea, not

00:01:38.780 --> 00:01:40.690
to find the area
under this curve,

00:01:40.690 --> 00:01:42.090
but to find the
volume if we were

00:01:42.090 --> 00:01:45.120
to rotate this curve
around the x-axis.

00:01:45.120 --> 00:01:48.900
So this is going to stretch our
powers of visualization here.

00:01:48.900 --> 00:01:50.620
So let's think
about what happens

00:01:50.620 --> 00:01:55.380
when we rotate this
thing around the x-axis.

00:01:55.380 --> 00:01:57.794
So if we were to rotate
it, and I'll look at it

00:01:57.794 --> 00:02:00.210
and say that we're looking it
a little bit from the right.

00:02:00.210 --> 00:02:03.890
So we get kind of a base that
looks something like this.

00:02:03.890 --> 00:02:06.940
So this is my best
attempt to draw it.

00:02:06.940 --> 00:02:10.990
So you have a base that
looks something like that.

00:02:10.990 --> 00:02:14.180
And then the rest
of the function,

00:02:14.180 --> 00:02:16.020
if we just think
about between 0 and 2,

00:02:16.020 --> 00:02:17.430
it looks like one
of those pieces

00:02:17.430 --> 00:02:20.370
from-- I don't know if you
ever played the game Sorry--

00:02:20.370 --> 00:02:23.910
or it looks like a little
bit of a weird hat.

00:02:23.910 --> 00:02:30.067
So it looks like this, and let
me shade it in a little bit

00:02:30.067 --> 00:02:31.400
so it looks something like that.

00:02:31.400 --> 00:02:32.816
And just so that
we're making sure

00:02:32.816 --> 00:02:35.590
we can visualize this
thing that's being rotated.

00:02:35.590 --> 00:02:38.250
We care about the entire
volume of the thing.

00:02:38.250 --> 00:02:40.720
Let me draw it from a
few different angles.

00:02:40.720 --> 00:02:45.290
So if I drew from the top, it
would look something like this.

00:02:45.290 --> 00:02:46.710
It'll become a
little more obvious

00:02:46.710 --> 00:02:48.560
that it looks
something like a hat.

00:02:48.560 --> 00:02:53.534
It would point up like this,
and it goes down like that.

00:02:53.534 --> 00:02:54.950
It would look
something like that.

00:02:54.950 --> 00:03:00.300
So in this angle, we're not
seeing the bottom of it.

00:03:00.300 --> 00:03:03.330
And if you were to
just orient yourself,

00:03:03.330 --> 00:03:06.060
the axes, in this
case, look like this.

00:03:06.060 --> 00:03:07.016
So this is the y-axis.

00:03:10.010 --> 00:03:13.640
And the x-axis goes right
inside of this thing

00:03:13.640 --> 00:03:16.650
and then pops out
the other side.

00:03:16.650 --> 00:03:18.640
And if this thing
was transparent,

00:03:18.640 --> 00:03:21.960
then you could
see the back side.

00:03:21.960 --> 00:03:23.890
It would look
something like that.

00:03:23.890 --> 00:03:26.500
The x-axis, if you
could see through it,

00:03:26.500 --> 00:03:28.979
would pop the base
right over there,

00:03:28.979 --> 00:03:31.020
would go right through
the base right over there.

00:03:31.020 --> 00:03:32.880
And it'd come out
on the other side.

00:03:32.880 --> 00:03:36.260
So this is one orientation
for the same thing.

00:03:36.260 --> 00:03:38.810
You could visualize it
from different angles.

00:03:38.810 --> 00:03:41.540
So let's think about how we
can take the volume of it.

00:03:41.540 --> 00:03:44.210
Well, instead of thinking
about the area of each

00:03:44.210 --> 00:03:46.850
of these rectangles, what
happens if we rotate each

00:03:46.850 --> 00:03:49.880
of these rectangles
around the x-axis?

00:03:49.880 --> 00:03:51.360
So let's do it.

00:03:51.360 --> 00:03:53.680
So let's take each of these.

00:03:53.680 --> 00:03:55.540
Let's say you have this
dx right over here,

00:03:55.540 --> 00:03:58.140
and you rotate it
around the x-axis.

00:03:58.140 --> 00:04:01.560
So if you were rotate this
thing around the x-axis--

00:04:01.560 --> 00:04:04.477
so I'm trying my best
to-- around the x-axis,

00:04:04.477 --> 00:04:05.060
you rotate it.

00:04:05.060 --> 00:04:06.090
What do you end up with?

00:04:08.600 --> 00:04:10.920
Well, you get something that
looks kind of like a coin,

00:04:10.920 --> 00:04:13.970
like a disk, like a
quarter of some kind.

00:04:13.970 --> 00:04:15.330
And let me draw it out here.

00:04:15.330 --> 00:04:18.320
So that same disk out here
would look something like this.

00:04:22.230 --> 00:04:27.860
And it has a depth of dx.

00:04:27.860 --> 00:04:31.170
So how can we find the
volume of that disk?

00:04:31.170 --> 00:04:33.130
Let me redraw it out here, too.

00:04:33.130 --> 00:04:35.740
It's really important to
visualize this stuff properly.

00:04:35.740 --> 00:04:39.160
So this is my x-axis.

00:04:39.160 --> 00:04:44.320
My disk looks
something like this.

00:04:44.320 --> 00:04:48.270
My best attempt at the x-axis
sits it right over there.

00:04:48.270 --> 00:04:49.870
It comes out of the center.

00:04:49.870 --> 00:04:55.040
And then this is the
surface of my disk.

00:04:55.040 --> 00:04:59.430
And then this right over
here is my depth dx.

00:04:59.430 --> 00:05:01.980
So that looks pretty good.

00:05:01.980 --> 00:05:03.890
And then let me just
shade it in a little bit

00:05:03.890 --> 00:05:05.740
to give you a little
bit of the depth.

00:05:05.740 --> 00:05:08.170
So how can we find
the volume of this?

00:05:08.170 --> 00:05:10.210
Well, like any disk
or cylinder, you just

00:05:10.210 --> 00:05:12.360
have to think about what
the area of this face

00:05:12.360 --> 00:05:14.100
is and then multiply
it times the depth.

00:05:14.100 --> 00:05:16.180
So what's the area of this base?

00:05:16.180 --> 00:05:18.260
Well, we know that
the area of a circle

00:05:18.260 --> 00:05:20.720
is equal to pi r squared.

00:05:20.720 --> 00:05:24.990
So if we know the
radius of this face,

00:05:24.990 --> 00:05:27.440
we can figure out
the area of the face.

00:05:27.440 --> 00:05:28.910
Well, what's the radius?

00:05:28.910 --> 00:05:30.390
Well, the radius
is just the height

00:05:30.390 --> 00:05:31.800
of that original rectangle.

00:05:31.800 --> 00:05:33.770
And for any x, the
height over here

00:05:33.770 --> 00:05:35.900
is going to be equal to f of x.

00:05:35.900 --> 00:05:38.830
And in this case, f
of x is x squared.

00:05:38.830 --> 00:05:41.625
So over here, our radius
is equal to x squared.

00:05:44.290 --> 00:05:48.380
So the area of the
face for a particular x

00:05:48.380 --> 00:05:55.940
is going to be equal to
pi times f of x squared.

00:05:55.940 --> 00:05:58.280
In this case, f
of x is x squared.

00:06:01.230 --> 00:06:02.710
Now, what's our
volume going to be?

00:06:02.710 --> 00:06:06.570
Well, our volume is going to be
our area times the depth here.

00:06:06.570 --> 00:06:10.530
It's going to be that
times the depth, times dx.

00:06:13.050 --> 00:06:18.810
So the volume of this thing
right over here-- so the volume

00:06:18.810 --> 00:06:23.420
just of this coin, I
guess you could call it,

00:06:23.420 --> 00:06:24.660
is going to be equal to.

00:06:24.660 --> 00:06:28.890
So my volume is going to
be equal to my area times

00:06:28.890 --> 00:06:36.270
dx, which is equal to pi
times x squared squared.

00:06:36.270 --> 00:06:39.310
So it's equal to pi--
x squared squared

00:06:39.310 --> 00:06:42.620
is x to the fourth--
pi x to the fourth dx.

00:06:46.630 --> 00:06:48.710
Now, this expression
right over here,

00:06:48.710 --> 00:06:52.470
this gave us the volume
just of one of those disks.

00:06:52.470 --> 00:06:55.060
But what we want is the
volume of this entire hat,

00:06:55.060 --> 00:06:58.899
or this entire bugle
or cone-looking,

00:06:58.899 --> 00:07:01.190
or I guess you could say the
front-of-a-trumpet-looking

00:07:01.190 --> 00:07:01.690
thing.

00:07:01.690 --> 00:07:02.930
So how could we do that?

00:07:02.930 --> 00:07:05.010
Well, the exact same technique.

00:07:05.010 --> 00:07:11.450
What happens if we were to take
the sum of all of these things?

00:07:11.450 --> 00:07:14.060
So let's do that, take the
sum of all of these things.

00:07:14.060 --> 00:07:17.622
And I'll switch to one color--
pi times x to the fourth dx.

00:07:17.622 --> 00:07:19.580
We're going take the sum
of all of these things

00:07:19.580 --> 00:07:21.550
from x is equal to 0 to 2.

00:07:21.550 --> 00:07:24.010
Those are the boundaries
that we started off with.

00:07:24.010 --> 00:07:25.510
I just defined them arbitrarily.

00:07:25.510 --> 00:07:27.980
We could do this, really,
for any two x values--

00:07:27.980 --> 00:07:30.100
between x is equal
to 0 and x equals 2.

00:07:30.100 --> 00:07:33.840
And we're going to take the
sum of the volumes of all

00:07:33.840 --> 00:07:35.030
of these coins.

00:07:35.030 --> 00:07:38.634
But the limit-- as the depths
get smaller and smaller

00:07:38.634 --> 00:07:40.550
and smaller and we have
more and more and more

00:07:40.550 --> 00:07:42.220
coins, at the limit,
we're actually

00:07:42.220 --> 00:07:45.460
going to get the volume
of our cone or our bugle

00:07:45.460 --> 00:07:47.540
or whatever we want to call it.

00:07:47.540 --> 00:07:49.550
So if we just evaluate
this definite integral,

00:07:49.550 --> 00:07:51.610
we have our volume.

00:07:51.610 --> 00:07:54.240
So let's see if we can do that.

00:07:54.240 --> 00:07:57.159
And now this is just taking
a standard definite integral.

00:07:57.159 --> 00:07:59.200
So this is going to be
equal to-- and I encourage

00:07:59.200 --> 00:08:01.380
you to try it out
before I do it.

00:08:01.380 --> 00:08:02.640
So we can take the pi out.

00:08:02.640 --> 00:08:05.130
So it's going to be equal
to pi times the integral

00:08:05.130 --> 00:08:08.780
from 0 to 2 of x
to the fourth dx.

00:08:08.780 --> 00:08:10.470
I don't like that color.

00:08:10.470 --> 00:08:12.680
Now, the antiderivative
of x to the fourth

00:08:12.680 --> 00:08:15.310
is x to the fifth over 5.

00:08:15.310 --> 00:08:21.720
So this is going to be equal to
pi times x to the fifth over 5.

00:08:21.720 --> 00:08:23.775
And we're going
to go from 0 to 2.

00:08:23.775 --> 00:08:26.770
So this is going to be
equal to pi times this thing

00:08:26.770 --> 00:08:27.440
evaluated at 2.

00:08:27.440 --> 00:08:27.940
Let's see.

00:08:27.940 --> 00:08:28.773
2 to the third is 8.

00:08:28.773 --> 00:08:29.970
2 to the fourth is 16.

00:08:29.970 --> 00:08:32.440
2 to the fifth is-- let
me just write it down.

00:08:32.440 --> 00:08:38.300
2 to the fifth over 5 minus
0 to the fifth over 5.

00:08:38.300 --> 00:08:39.799
And this is going
to be equal to-- 2

00:08:39.799 --> 00:08:44.700
to the fifth is 32, so it's
going to be equal to pi times

00:08:44.700 --> 00:08:48.690
32/5 minus-- well,
this is just 0--

00:08:48.690 --> 00:08:54.790
minus 0, which is
equal to 32 pi over 5.

00:08:54.790 --> 00:08:55.480
And we're done.

00:08:55.480 --> 00:08:57.540
We were able to
figure out the volume

00:08:57.540 --> 00:09:00.240
of this kind of wacky shape.


Les Miserables and France's many revolutions

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:02.750
There's a very popular, and
I thought, good movie out,

00:00:02.750 --> 00:00:05.720
based on the play, which
is based on the novel

00:00:05.720 --> 00:00:07.872
by Victor Hugo, Les Miserables.

00:00:07.872 --> 00:00:09.330
And I'm sure I'm
mispronouncing it.

00:00:09.330 --> 00:00:12.118
It means "the
miserable," for those

00:00:12.118 --> 00:00:13.747
for whom it was not obvious.

00:00:13.747 --> 00:00:15.830
And what I want to give
is a little bit of context

00:00:15.830 --> 00:00:19.990
because in that
book, play, movie

00:00:19.990 --> 00:00:21.910
there is an
attempted revolution.

00:00:21.910 --> 00:00:23.800
And sometimes people
try to associate it

00:00:23.800 --> 00:00:25.510
with the French Revolution.

00:00:25.510 --> 00:00:28.900
But that is not depicting
the French Revolution

00:00:28.900 --> 00:00:30.560
that people talk
about when they talk

00:00:30.560 --> 00:00:31.910
about the French Revolution.

00:00:31.910 --> 00:00:34.230
And so I thought I would
give a little bit of scaffold

00:00:34.230 --> 00:00:37.400
of French history in the
late 1700s and all the way

00:00:37.400 --> 00:00:40.570
through the mid-1800s to
give a little bit of context

00:00:40.570 --> 00:00:41.930
on the matter.

00:00:41.930 --> 00:00:46.760
So let's start off in 1789.

00:00:46.760 --> 00:00:49.520
That's when you have
the French Revolution.

00:00:49.520 --> 00:00:51.240
The First French
Revolution, we could

00:00:51.240 --> 00:00:54.522
say, or the French Revolution,
which is the French Revolution

00:00:54.522 --> 00:00:55.980
that people talk
about when they're

00:00:55.980 --> 00:00:57.780
talking about the
French Revolution.

00:00:57.780 --> 00:01:02.670
It was all about deposing
Louis XVI and his wife, Maria

00:01:02.670 --> 00:01:03.170
Antoinette.

00:01:03.170 --> 00:01:04.530
This is her body there.

00:01:04.530 --> 00:01:06.252
I think she just
got guillotined.

00:01:06.252 --> 00:01:06.960
This is her head.

00:01:06.960 --> 00:01:08.520
It was very bloody revolution.

00:01:08.520 --> 00:01:13.700
This is the storming of the
Bastille right over here.

00:01:13.700 --> 00:01:18.500
And that starts the beginning
of the First Republic in France.

00:01:18.500 --> 00:01:19.840
The First Republic.

00:01:19.840 --> 00:01:22.830
So there were all these
dreams and aspirations

00:01:22.830 --> 00:01:25.740
that France would now be
a country of the people.

00:01:25.740 --> 00:01:28.960
Not too dissimilar
to the United States.

00:01:28.960 --> 00:01:33.780
But revolutions are not so
easy, or so clean, or so fast.

00:01:33.780 --> 00:01:39.380
And France had to go through
a long period of pain

00:01:39.380 --> 00:01:42.960
before it could really establish
itself as a real republic.

00:01:42.960 --> 00:01:45.961
But let's keep going on
further off in history.

00:01:45.961 --> 00:01:47.460
Let me do that in
a different color.

00:01:47.460 --> 00:01:49.650
I'll do the timeline in white.

00:01:49.650 --> 00:01:51.610
So let's fast forward.

00:01:51.610 --> 00:01:57.530
Let's fast forward to 1799.

00:01:57.530 --> 00:02:01.200
This is when Napoleon
Bonaparte comes to power.

00:02:01.200 --> 00:02:02.811
So when people talk
about Napoleon,

00:02:02.811 --> 00:02:04.560
they are talking about
Napoleon Bonaparte.

00:02:04.560 --> 00:02:06.030
We'll see that there
are other Napoleons.

00:02:06.030 --> 00:02:08.490
But if people just say, hey,
Napoleon did this or that,

00:02:08.490 --> 00:02:10.520
they're talking about
Napoleon Bonaparte.

00:02:10.520 --> 00:02:13.400
So this is, Napoleon
comes to power.

00:02:13.400 --> 00:02:14.530
Napoleon.

00:02:14.530 --> 00:02:17.410
And he officially ends
the First Republic in 1804

00:02:17.410 --> 00:02:20.640
because he declares
himself emperor.

00:02:20.640 --> 00:02:21.722
But let's fast forward.

00:02:21.722 --> 00:02:23.680
There's many videos on
the Khan Academy dealing

00:02:23.680 --> 00:02:26.390
with the Napoleonic Wars
and the French Revolution.

00:02:26.390 --> 00:02:29.560
But let's fast forward to 1815.

00:02:29.560 --> 00:02:31.790
So let me do that
in white again.

00:02:31.790 --> 00:02:33.289
So you get to 1815.

00:02:33.289 --> 00:02:33.830
So let's see.

00:02:33.830 --> 00:02:34.860
That was about 10 years.

00:02:34.860 --> 00:02:36.860
I need to go about 16 years.

00:02:36.860 --> 00:02:40.330
So that would put us at
1815 right over there.

00:02:40.330 --> 00:02:47.310
1815 is, essentially,
Napoleon faces

00:02:47.310 --> 00:02:49.470
his Waterloo, which was
literally at Waterloo.

00:02:49.470 --> 00:02:52.810
That's why people talk
about facing your Waterloo.

00:02:52.810 --> 00:02:56.160
So he was banished for
a little bit to Elba.

00:02:56.160 --> 00:02:57.260
He was able to come back.

00:02:57.260 --> 00:02:59.900
He had 100, actually
111 days in power.

00:02:59.900 --> 00:03:02.270
But then he was
finally defeated.

00:03:02.270 --> 00:03:05.840
And then he was
finally put into exile

00:03:05.840 --> 00:03:07.910
at Saint Helena, where he died.

00:03:07.910 --> 00:03:11.460
In 1815, you essentially
have the restoration

00:03:11.460 --> 00:03:12.860
of the Bourbon monarchy.

00:03:12.860 --> 00:03:19.160
So Louis XVI's younger
brother comes to power.

00:03:19.160 --> 00:03:23.730
And they call him Louis XVIII.

00:03:23.730 --> 00:03:28.560
So this is Louis XVIII, which
raises a very good question--

00:03:28.560 --> 00:03:30.440
What happened to Louis XVII?

00:03:30.440 --> 00:03:33.520
Louis XVII was
Louis XVI's son, who

00:03:33.520 --> 00:03:40.380
died in prison at the age of 10
in 1795 during the Revolution

00:03:40.380 --> 00:03:42.770
or during the revolutionary
period, I guess we could say.

00:03:42.770 --> 00:03:44.370
So this right over here.

00:03:44.370 --> 00:03:45.660
This right over here.

00:03:45.660 --> 00:03:47.035
Napoleon ends -- let me draw.

00:03:47.035 --> 00:03:50.930
This is Napoleon's-- let me do
this in the same color I did

00:03:50.930 --> 00:03:52.430
Napoleon in.

00:03:52.430 --> 00:03:58.295
So 1799 to, actually, 1814 is
when Napoleon's reign ended

00:03:58.295 --> 00:03:59.920
but then he came back
for a little bit.

00:03:59.920 --> 00:04:02.270
So I'll draw a little bit
of a dotted line here.

00:04:02.270 --> 00:04:03.820
A little dotted line.

00:04:03.820 --> 00:04:07.270
And then 1814 was the
formal restoration

00:04:07.270 --> 00:04:09.312
of the Bourbon monarchy.

00:04:09.312 --> 00:04:11.520
And, but of course, Napoleon
comes back a little bit.

00:04:11.520 --> 00:04:16.029
But after Waterloo, it's
really firmly established.

00:04:16.029 --> 00:04:18.269
So you have the restoration
of the Bourbon monarchy.

00:04:18.269 --> 00:04:21.709
So this is Louis the XVIII.

00:04:21.709 --> 00:04:24.150
And then in 1824 he dies.

00:04:24.150 --> 00:04:25.095
And he dies childless.

00:04:27.900 --> 00:04:28.534
1824.

00:04:28.534 --> 00:04:29.075
So let's see.

00:04:29.075 --> 00:04:30.080
It's about nine years.

00:04:30.080 --> 00:04:31.960
So that will put us
about right over there.

00:04:31.960 --> 00:04:34.160
1824.

00:04:34.160 --> 00:04:35.350
He dies childless.

00:04:35.350 --> 00:04:40.220
And so his younger brother,
Charles X comes to power.

00:04:40.220 --> 00:04:43.560
So then you have
Charles X. Charles.

00:04:43.560 --> 00:04:47.960
I'll do all the
Bourbons in purple.

00:04:47.960 --> 00:04:52.830
Charles X. And this is
Charles X right over here.

00:04:52.830 --> 00:04:57.370
And so let's go a little bit
further off into history.

00:04:57.370 --> 00:05:01.620
You fast forward
all the way to 1830.

00:05:01.620 --> 00:05:05.370
You fast forward to 1830.

00:05:05.370 --> 00:05:06.740
A lot of discontent.

00:05:06.740 --> 00:05:09.642
And now you have the
Second French Revolution.

00:05:09.642 --> 00:05:11.100
You might say, oh
hey, this must be

00:05:11.100 --> 00:05:12.930
what Les Miserables
is all about.

00:05:12.930 --> 00:05:13.990
No, we're not there yet.

00:05:13.990 --> 00:05:17.730
Les Mis is not about the
Second French Revolution,

00:05:17.730 --> 00:05:20.250
sometimes called
the July Revolution.

00:05:20.250 --> 00:05:22.570
July Revolution.

00:05:22.570 --> 00:05:25.330
And this revolution actually
did not-- it was successful--

00:05:25.330 --> 00:05:27.870
but it did not establish
a republic at this point.

00:05:27.870 --> 00:05:30.620
It instead installed--
and this whole time there

00:05:30.620 --> 00:05:32.320
was a liberalization.

00:05:32.320 --> 00:05:34.300
The monarchy, even when
it was put in power,

00:05:34.300 --> 00:05:37.420
had a gradual decline in
how much power it had.

00:05:37.420 --> 00:05:42.950
But after the July Revolution,
they put in Charles X's cousin.

00:05:42.950 --> 00:05:45.486
Charles X's cousin who
was the Duke of Orleans.

00:05:48.700 --> 00:05:52.820
Who was this guy, this
guy right over here.

00:05:52.820 --> 00:05:56.490
Louis Philippe I. So
let me write that.

00:05:56.490 --> 00:06:06.490
So this is Louis Philippe I.

00:06:06.490 --> 00:06:08.340
And so you're saying,
Sal, you started off

00:06:08.340 --> 00:06:10.290
this video talking
about Les Mis.

00:06:10.290 --> 00:06:12.480
You haven't mentioned
Les Mis yet.

00:06:12.480 --> 00:06:14.010
Give me a little bit of context.

00:06:14.010 --> 00:06:15.760
So now I will give you context.

00:06:15.760 --> 00:06:18.740
So if you watch the movie, it
starts off with Jean Valjean.

00:06:18.740 --> 00:06:22.660
He's at a shipping
place where they're

00:06:22.660 --> 00:06:24.330
repairing ships of some sort.

00:06:24.330 --> 00:06:27.260
That was in 1815 after Waterloo.

00:06:27.260 --> 00:06:32.020
So it was under
Louis XVIII's regime.

00:06:32.020 --> 00:06:33.370
So that's right over here.

00:06:33.370 --> 00:06:35.036
Let me see where the
start of the movie.

00:06:35.036 --> 00:06:40.640
The start of the movie
is right about there.

00:06:40.640 --> 00:06:44.530
And then the real climax of the
movie, which is this rebellion.

00:06:44.530 --> 00:06:49.100
There's these barricades
being set up in Paris.

00:06:49.100 --> 00:06:51.050
You have all these
young idealistic folks

00:06:51.050 --> 00:06:53.480
who are trying to
overthrow the government.

00:06:53.480 --> 00:06:56.640
This does not happen until 1832.

00:06:56.640 --> 00:06:59.050
This is 1832.

00:06:59.050 --> 00:07:00.360
I'll do it right over here.

00:07:00.360 --> 00:07:03.250
1832.

00:07:03.250 --> 00:07:04.940
And what catalyzed
that-- there were

00:07:04.940 --> 00:07:06.742
several things that
was catalyzing that.

00:07:06.742 --> 00:07:08.200
And, actually, most
revolutions are

00:07:08.200 --> 00:07:10.420
catalyzed by just
economic discontent.

00:07:10.420 --> 00:07:12.440
If people are rich, and
happy, and have jobs,

00:07:12.440 --> 00:07:14.000
and aren't getting
sick, most people

00:07:14.000 --> 00:07:15.930
aren't in the mood to revolt.

00:07:15.930 --> 00:07:19.040
But in 1832, as
you could imagine,

00:07:19.040 --> 00:07:21.060
the economic situation
was not good.

00:07:21.060 --> 00:07:25.800
There was also a very
nasty outbreak of cholera.

00:07:25.800 --> 00:07:29.710
And what really catalyzed
the events in Les Mis,

00:07:29.710 --> 00:07:32.220
and they even refer
to it in the movie,

00:07:32.220 --> 00:07:36.260
is the death of this
chap right over here.

00:07:36.260 --> 00:07:38.180
Jean Maximilien Lamarque.

00:07:38.180 --> 00:07:40.300
Let me write the name down.

00:07:40.300 --> 00:07:45.720
Jean Maximilien--
and I'm sure I'm

00:07:45.720 --> 00:07:49.460
mispronouncing it-- Lamarque.

00:07:49.460 --> 00:07:55.730
And he dies in June of 1832.

00:07:55.730 --> 00:07:58.980
And he was very sympathetic
to the plight of the poor,

00:07:58.980 --> 00:08:01.590
to the plight of the common man.

00:08:01.590 --> 00:08:05.630
And the average folks
said, hey look, he's

00:08:05.630 --> 00:08:06.770
our guy in government.

00:08:06.770 --> 00:08:09.740
And he had an influential
role in government.

00:08:09.740 --> 00:08:12.080
When he died, they
were like, look,

00:08:12.080 --> 00:08:16.150
we don't have anybody else
in a high position who

00:08:16.150 --> 00:08:17.640
can speak for us.

00:08:17.640 --> 00:08:21.570
Let's use his funeral as
a catalyst for revolt.

00:08:21.570 --> 00:08:25.210
And you saw that happening
in the movie Les Mis.

00:08:25.210 --> 00:08:27.540
So Les Mis, that
climactic moment,

00:08:27.540 --> 00:08:31.470
that is the June
Rebellion of 1832.

00:08:31.470 --> 00:08:34.919
And it's not-- you don't have
to have amazing comprehension

00:08:34.919 --> 00:08:38.730
of watching movies to realize
that this was unsuccessful.

00:08:38.730 --> 00:08:41.610
So this right here, didn't work.

00:08:41.610 --> 00:08:42.900
Didn't work.

00:08:42.900 --> 00:08:44.520
If it did work,
It might have been

00:08:44.520 --> 00:08:46.940
called the Third
French Revolution.

00:08:46.940 --> 00:08:48.040
But it was not.

00:08:48.040 --> 00:08:49.790
It was an unsuccessful
revolution

00:08:49.790 --> 00:08:52.310
or unsuccessful
rebellion, really.

00:08:52.310 --> 00:08:54.220
And it was that Victor
Hugo observed it.

00:08:54.220 --> 00:08:57.120
And that's why he's able to
recount it in so much detail.

00:08:57.120 --> 00:08:58.720
The barricading,
the young people,

00:08:58.720 --> 00:09:01.170
the shooting in the
streets, all the rest.

00:09:01.170 --> 00:09:03.130
So this is a little
bit of review.

00:09:03.130 --> 00:09:05.530
When people talk about
the French Revolution,

00:09:05.530 --> 00:09:08.420
they're usually talking
about the French Revolution.

00:09:08.420 --> 00:09:10.650
1789.

00:09:10.650 --> 00:09:13.600
Begins to establish
the First Republic.

00:09:13.600 --> 00:09:15.760
It was a successful revolution.

00:09:15.760 --> 00:09:18.150
The Second French Revolution.

00:09:18.150 --> 00:09:20.220
This is the July Revolution.

00:09:20.220 --> 00:09:23.090
This is in 1830.

00:09:23.090 --> 00:09:26.880
This puts into power Louis
Philippe I, the person

00:09:26.880 --> 00:09:29.810
that they're trying to
overthrow in Les Mis.

00:09:29.810 --> 00:09:33.040
And they don't establish the
Second Republic after this.

00:09:33.040 --> 00:09:37.260
For the Second Republic,
we have to go all the way

00:09:37.260 --> 00:09:42.240
to the revolution
of-- let me make

00:09:42.240 --> 00:09:44.560
sure I can scroll properly.

00:09:44.560 --> 00:09:45.570
Go all the way.

00:09:45.570 --> 00:09:48.000
So let me continue my timeline.

00:09:48.000 --> 00:09:52.280
So this is Louis
Philippe to go 18 years.

00:09:52.280 --> 00:10:06.020
18 years to 1848 where you have
the Third French Revolution,

00:10:06.020 --> 00:10:09.630
which leads to the popular
election of Napoleon

00:10:09.630 --> 00:10:12.230
Bonaparte's nephew.

00:10:12.230 --> 00:10:14.570
Napoleon Bonaparte's nephew.

00:10:14.570 --> 00:10:17.080
Louis Napoleon Bonaparte.

00:10:17.080 --> 00:10:19.820
This guy right over here.

00:10:19.820 --> 00:10:22.820
But France is still not done.

00:10:22.820 --> 00:10:26.940
It still can't establish itself
as a long-lasting republic.

00:10:26.940 --> 00:10:30.890
In 1851, this character
declares himself emperor.

00:10:30.890 --> 00:10:34.730
So 1851, he too declares
himself emperor.

00:10:34.730 --> 00:10:39.260
And France is not finally
freed of kings and emperors

00:10:39.260 --> 00:10:43.590
until 1870.

00:10:43.590 --> 00:10:49.260
So let's do this run all the
way to 1870 where France,

00:10:49.260 --> 00:10:52.030
essentially, loses the
Franco-Prussian War.

00:10:52.030 --> 00:10:55.220
And this character, this
character right over here,

00:10:55.220 --> 00:10:56.570
is deposed.

00:10:56.570 --> 00:11:01.660
And you have the establishment
of the Third French Republic.


U-substitution with definite integral

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.870
We now have a definite
integral going

00:00:02.870 --> 00:00:05.790
from x equals 0 to x
equals square root of pi,

00:00:05.790 --> 00:00:10.080
of x sine of x squared dx.

00:00:10.080 --> 00:00:12.730
So the first thing that you
might want to ask yourself

00:00:12.730 --> 00:00:16.300
is whether u substitution
is appropriate here.

00:00:16.300 --> 00:00:19.790
And if it is, what would
you set u to be equal to?

00:00:19.790 --> 00:00:23.490
I'll let you think
about that for a second.

00:00:23.490 --> 00:00:26.400
Well, we see here this
expression, x squared.

00:00:26.400 --> 00:00:28.990
And if we were to just take
the derivative of x squared,

00:00:28.990 --> 00:00:29.910
it would be 2x.

00:00:29.910 --> 00:00:32.430
We have an x here,
so it's almost a 2x.

00:00:32.430 --> 00:00:35.930
To turn this x into 2x we
could multiply it by 2.

00:00:35.930 --> 00:00:38.150
But we can't just
only multiply it by 2,

00:00:38.150 --> 00:00:39.620
because that would
change the value

00:00:39.620 --> 00:00:41.110
of this entire expression.

00:00:41.110 --> 00:00:43.290
We'd also have to
multiply by 1/2.

00:00:43.290 --> 00:00:46.550
So all I've done so far is I
multiplied by 2 and by 1/2,

00:00:46.550 --> 00:00:48.980
which is the same thing
as multiplying by 1.

00:00:48.980 --> 00:00:53.480
I have not changed the value of
this entire definite integral.

00:00:53.480 --> 00:00:56.470
Now, let's do our
u substitution.

00:00:56.470 --> 00:01:02.640
If we set u as
equal to x squared,

00:01:02.640 --> 00:01:07.545
we now know that du is
going to be equal to 2xdx.

00:01:11.040 --> 00:01:12.910
And so, what can we see here?

00:01:12.910 --> 00:01:15.330
Well, we have our u there,
that's how we defined it.

00:01:15.330 --> 00:01:20.700
And our du is going
to be 2x times dx.

00:01:20.700 --> 00:01:24.040
So this entire
inside, or the thing

00:01:24.040 --> 00:01:27.080
that we're taking the integral
of, the definite integral of,

00:01:27.080 --> 00:01:35.830
is going to be-- we can rewrite
it as sine of x squared.

00:01:35.830 --> 00:01:39.080
But x squared is
just u, so sine of u.

00:01:39.080 --> 00:01:44.320
And instead of 2xdx, we
know that 2xdx is now du.

00:01:44.320 --> 00:01:46.820
So the thing that we're taking
the integral of in terms of u

00:01:46.820 --> 00:01:48.940
could be expressed in this way.

00:01:48.940 --> 00:01:51.440
And so we have our integral.

00:01:51.440 --> 00:01:53.650
But these boundaries
of integration

00:01:53.650 --> 00:01:55.430
were written in terms of x.

00:01:55.430 --> 00:01:57.570
We could have written
this as x is equal to 0,

00:01:57.570 --> 00:02:00.150
to x is equal to
square root of pi.

00:02:00.150 --> 00:02:03.286
Now we want to write
them in terms of u.

00:02:03.286 --> 00:02:07.210
And let me not forget that I
had this 1/2 sitting out here.

00:02:07.210 --> 00:02:10.449
So how do we write
it in terms of u?

00:02:10.449 --> 00:02:14.690
So when x is equal to 0, what
is u going to be equal to?

00:02:14.690 --> 00:02:16.510
This boundary is
when x equals 0.

00:02:16.510 --> 00:02:21.100
Well, when x is equal to
0, u is also equal to 0.

00:02:21.100 --> 00:02:24.100
So this is from u equals zero,
this is the same boundary.

00:02:24.100 --> 00:02:26.030
When x is equal to
0, u is equal to 0,

00:02:26.030 --> 00:02:28.060
based on how we defined u.

00:02:28.060 --> 00:02:29.260
Now what about here?

00:02:29.260 --> 00:02:34.690
What is u equal to when x is
equal to the square root of pi?

00:02:34.690 --> 00:02:36.470
Well, u is equal to
the square root of pi

00:02:36.470 --> 00:02:38.520
squared, let me write this down.

00:02:38.520 --> 00:02:45.400
u is going to be equal to the
square root of pi squared, when

00:02:45.400 --> 00:02:49.950
x is equal to the square root of
pi, which is just equal to pi.

00:02:49.950 --> 00:02:56.640
So this is going to be from
u equals 0, to u equals pi.

00:02:56.640 --> 00:02:58.900
I didn't have to explicitly
write this u equals here,

00:02:58.900 --> 00:03:00.525
but that makes it a
little bit clearer.

00:03:00.525 --> 00:03:03.673
Our boundaries are now
written in terms of u.

00:03:03.673 --> 00:03:08.740
And so we are now
ready to evaluate

00:03:08.740 --> 00:03:10.570
this indefinite integral.

00:03:10.570 --> 00:03:17.410
So this whole thing is going
to be equal to 1/2 times

00:03:17.410 --> 00:03:18.190
the expression.

00:03:18.190 --> 00:03:21.760
Well, what's the
antiderivative of sine of u?

00:03:21.760 --> 00:03:24.160
Well, the derivative
of cosine of u,

00:03:24.160 --> 00:03:25.680
let me write it over here.

00:03:25.680 --> 00:03:27.900
ddx-- or maybe I should
write in terms of u--

00:03:27.900 --> 00:03:35.370
d/du cosine u is equal to
negative sine, negative sine

00:03:35.370 --> 00:03:36.220
of u.

00:03:36.220 --> 00:03:40.960
So the derivative of
negative cosine of u

00:03:40.960 --> 00:03:44.790
is equal to sine of u.

00:03:44.790 --> 00:03:48.086
So the antiderivative of sine
of u is negative cosine of u.

00:03:52.780 --> 00:03:58.500
And we're going to evaluate
that from, at u equals pi.

00:03:58.500 --> 00:04:01.510
And from that we're going to
subtract it, evaluate it at u

00:04:01.510 --> 00:04:03.130
equals 0.

00:04:03.130 --> 00:04:04.930
u equals 0.

00:04:04.930 --> 00:04:09.550
So this is going to be
equal to 1/2 times--

00:04:09.550 --> 00:04:12.190
well, what happens when we
evaluate it at u equals pi?

00:04:12.190 --> 00:04:15.410
Negative cosine of pi.

00:04:15.410 --> 00:04:17.579
Cosine of pi, if we
just-- and I always

00:04:17.579 --> 00:04:22.040
have to redraw my unit circles
here-- so cosine of pi.

00:04:22.040 --> 00:04:23.780
We're at this point
on the unit circle.

00:04:23.780 --> 00:04:25.180
Cosine is the x-coordinate.

00:04:25.180 --> 00:04:28.460
Cosine of pi is-- let
me just write it down--

00:04:28.460 --> 00:04:32.490
it's going to be
negative cosine of pi.

00:04:32.490 --> 00:04:35.180
And then from that, we
are going to subtract

00:04:35.180 --> 00:04:37.350
this evaluated when u equals 0.

00:04:37.350 --> 00:04:46.740
So minus negative cosine of 0.

00:04:46.740 --> 00:04:48.640
Not cosine of theta.

00:04:48.640 --> 00:04:50.280
That's what it's
going to be equal to.

00:04:50.280 --> 00:04:52.180
And now we're ready
to evaluate it.

00:04:52.180 --> 00:04:54.502
Cosine of pi is -1.

00:04:54.502 --> 00:04:56.210
But then we have this
negative out front,

00:04:56.210 --> 00:04:59.310
so this whole thing
will evaluate to 1.

00:04:59.310 --> 00:05:00.940
Cosine of 0.

00:05:00.940 --> 00:05:04.010
Cosine of 0 is 1.

00:05:04.010 --> 00:05:06.460
And then we're
subtracting a -1, which

00:05:06.460 --> 00:05:08.190
is the exact same
thing as adding a 1.

00:05:08.190 --> 00:05:12.360
So this whole thing
also becomes a 1.

00:05:12.360 --> 00:05:16.210
And so in here we have
1 plus 1 times 1/2.

00:05:16.210 --> 00:05:19.540
So this whole thing simplifies
to-- ready for our drum

00:05:19.540 --> 00:05:21.590
roll-- 1.

00:05:21.590 --> 00:05:23.600
And you see, when we
did u substitution

00:05:23.600 --> 00:05:25.250
with a definite
integral, as long

00:05:25.250 --> 00:05:27.790
as we changed our
boundaries in terms of u,

00:05:27.790 --> 00:05:30.500
we didn't have to go and
then undo the substitution.

00:05:30.500 --> 00:05:32.760
We didn't even have to go
and convert back into x.

00:05:32.760 --> 00:05:35.020
We were able to evaluate
the definite integral

00:05:35.020 --> 00:05:37.860
in terms of u.


U-substitution and back substitution

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.540
We're faced with the indefinite
integral of x plus 3,

00:00:03.540 --> 00:00:06.970
times x minus 1 to the fifth dx.

00:00:06.970 --> 00:00:09.870
Now, we could solve this by
literally multiplying out

00:00:09.870 --> 00:00:11.430
what x minus 1 to the fifth is.

00:00:11.430 --> 00:00:12.980
Maybe using the
binomial theorem,

00:00:12.980 --> 00:00:13.980
that would take a while.

00:00:13.980 --> 00:00:15.920
And then we'd multiply
that times x plus 3,

00:00:15.920 --> 00:00:17.820
and we'd end up with
some polynomial.

00:00:17.820 --> 00:00:20.150
And we could take the
antiderivative that way.

00:00:20.150 --> 00:00:22.780
Or we could maybe make
a substitution here

00:00:22.780 --> 00:00:25.840
that could simplify
this expression here.

00:00:25.840 --> 00:00:27.810
Make it something that's
a little bit easier

00:00:27.810 --> 00:00:30.020
to take the antiderivative of.

00:00:30.020 --> 00:00:32.439
And this isn't going to be
kind of the more traditional u

00:00:32.439 --> 00:00:34.620
substitution, where we just
set u equal to something

00:00:34.620 --> 00:00:36.280
and see if its
derivative is there.

00:00:36.280 --> 00:00:38.490
But it's a kind of a form
of u substitution, where

00:00:38.490 --> 00:00:41.420
we do set u equal to something
and see if it simplifies

00:00:41.420 --> 00:00:44.560
our expression in a way
that, I guess, simplifies it.

00:00:44.560 --> 00:00:46.550
So let's try things out.

00:00:46.550 --> 00:00:48.960
So we have this x
minus 1 to the fifth.

00:00:48.960 --> 00:00:50.680
That would be a
pain to expand out.

00:00:50.680 --> 00:00:53.560
It would be nice if this
was just a u to the fifth.

00:00:53.560 --> 00:00:57.390
So let's just set this, let's
just set this to be equal to u.

00:00:57.390 --> 00:01:01.370
So let's set u as
equal to x minus 1.

00:01:01.370 --> 00:01:05.850
And in that case,
du is equal to dx.

00:01:05.850 --> 00:01:09.590
We could write
du/dx is equal to 1,

00:01:09.590 --> 00:01:12.890
derivative of x, derivative
of negative 1 is just 0.

00:01:12.890 --> 00:01:17.320
And these two are completely
equivalent statements.

00:01:17.320 --> 00:01:18.810
And so if we did
that, how could we

00:01:18.810 --> 00:01:21.440
rewrite this entire expression?

00:01:21.440 --> 00:01:26.140
Well, it would be equal
to the integral of-- well,

00:01:26.140 --> 00:01:29.633
we have x plus 3 right over
here, this is neither just u

00:01:29.633 --> 00:01:32.940
nor is it du.

00:01:32.940 --> 00:01:35.300
So let's think about
how, what we can do here.

00:01:35.300 --> 00:01:40.720
Well, we could say, if
u is equal to x minus 1,

00:01:40.720 --> 00:01:44.140
we could add 1 to both
sides of this equation.

00:01:44.140 --> 00:01:49.230
And we could say u
plus 1 is equal to x.

00:01:49.230 --> 00:01:52.310
And so for x, we can
substitute that with u plus 1.

00:01:52.310 --> 00:01:53.500
So let's do that.

00:01:53.500 --> 00:01:57.010
So we're kind of back
substituting in for x.

00:01:57.010 --> 00:02:00.800
So x is equal to u plus 1.

00:02:00.800 --> 00:02:02.800
And I'm just trying to
see if I can do something

00:02:02.800 --> 00:02:04.230
to simplify this expression.

00:02:04.230 --> 00:02:10.620
So x is u plus 1, then
we have our plus 3 there,

00:02:10.620 --> 00:02:13.200
times x minus 1 to the fifth.

00:02:13.200 --> 00:02:15.650
x minus 1 was u, that's
the simplification

00:02:15.650 --> 00:02:16.770
we wanted to make.

00:02:16.770 --> 00:02:20.250
So times u to the fifth power.

00:02:20.250 --> 00:02:22.750
And dx is the same thing as du.

00:02:22.750 --> 00:02:25.450
So du.

00:02:25.450 --> 00:02:27.250
Now did this get us anywhere?

00:02:27.250 --> 00:02:29.350
Can we simplify this
to a form that it's

00:02:29.350 --> 00:02:31.890
easy to take the
antiderivative of?

00:02:31.890 --> 00:02:33.370
Well, it looks like it did.

00:02:33.370 --> 00:02:33.940
Let's see.

00:02:33.940 --> 00:02:37.850
We can rewrite this as this
expression right over here

00:02:37.850 --> 00:02:39.040
is just u plus 4.

00:02:42.520 --> 00:02:48.730
Times u to the fifth-- I'll do
it all in one color, now-- du.

00:02:48.730 --> 00:02:51.980
And the reason why this
simplified things-- the way

00:02:51.980 --> 00:02:54.700
this simplified things
is taking x minus one

00:02:54.700 --> 00:02:56.950
to the fifth, that'd be a
really hard thing to expand.

00:02:56.950 --> 00:02:59.230
But u to the fifth is
just u to the fifth.

00:02:59.230 --> 00:03:01.040
And then it just
changed this x plus 3

00:03:01.040 --> 00:03:03.460
into a u plus 4, which is
still a pretty straightforward

00:03:03.460 --> 00:03:04.320
expression.

00:03:04.320 --> 00:03:08.580
And now we can just
distribute the u to the fifth.

00:03:08.580 --> 00:03:15.250
So we are left with u to
the sixth power plus 4,

00:03:15.250 --> 00:03:17.670
u to the fifth du.

00:03:17.670 --> 00:03:19.440
And this is a pretty
straightforward thing

00:03:19.440 --> 00:03:20.846
to take the antiderivative of.

00:03:20.846 --> 00:03:22.220
Now you might be
saying, hey Sal,

00:03:22.220 --> 00:03:25.630
this was-- how did you know
to set u equal to be that.

00:03:25.630 --> 00:03:27.230
And oftentimes with
integration, it's

00:03:27.230 --> 00:03:29.146
going to take a little
bit of trial and error.

00:03:29.146 --> 00:03:30.740
There's a certain
bit of an art to it.

00:03:30.740 --> 00:03:33.327
But here the realization was
well, x minus 1 to the fifth

00:03:33.327 --> 00:03:34.410
can be really complicated.

00:03:34.410 --> 00:03:36.743
Maybe u to the fifth might
make it a little bit simpler.

00:03:36.743 --> 00:03:38.590
And that did just
happen to work.

00:03:38.590 --> 00:03:40.704
You could have tried u
is equal to x plus 3.

00:03:40.704 --> 00:03:42.120
But it wouldn't
have simplified it

00:03:42.120 --> 00:03:45.560
as nicely as u equal
to x minus 1 did.

00:03:45.560 --> 00:03:48.940
But let's finish with this
integral right over here.

00:03:48.940 --> 00:03:51.730
So this is going to be equal
to the antiderivative of u

00:03:51.730 --> 00:03:52.780
to the sixth.

00:03:52.780 --> 00:03:56.970
Well, that's just u to
the seventh over seven.

00:03:56.970 --> 00:03:59.530
Plus the antiderivative
of u to the fifth, that's

00:03:59.530 --> 00:04:01.160
u to the sixth over six.

00:04:01.160 --> 00:04:02.980
But we have the 4
out here, so it's

00:04:02.980 --> 00:04:07.650
4 times u to the sixth over six.

00:04:07.650 --> 00:04:12.180
And then we have a plus
C. And this, the 4u 6

00:04:12.180 --> 00:04:15.790
is the same thing as 2/3, so we
can rewrite this whole thing as

00:04:15.790 --> 00:04:21.899
equal to u to the seventh over
seven, plus 2/3 u to the sixth,

00:04:21.899 --> 00:04:23.020
plus C.

00:04:23.020 --> 00:04:25.910
And now we just have to
undo our u substitution.

00:04:25.910 --> 00:04:28.160
u is equal to x minus 1.

00:04:28.160 --> 00:04:35.330
So this is going to be equal to
x minus 1 to the seventh over 7

00:04:35.330 --> 00:04:43.110
plus 2/3 times x minus
1 to the sixth, plus C.

00:04:43.110 --> 00:04:44.600
And we're all done.

00:04:44.600 --> 00:04:48.450
We were able to take a
fairly hairy-- or what

00:04:48.450 --> 00:04:51.200
could have been a hairy thing
if we had to expand this out--

00:04:51.200 --> 00:04:54.730
and we were able to take
the antiderivative by doing

00:04:54.730 --> 00:04:57.110
a little bit of
this u substitution

00:04:57.110 --> 00:04:59.600
and u back substitution.


Doing u-substitution twice (second time with w)

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:05.170
Let's see if we can take the
integral of cosine of 5x over e

00:00:05.170 --> 00:00:07.540
to the sine of 5x dx.

00:00:07.540 --> 00:00:10.340
And there's a crow squawking
outside of my window,

00:00:10.340 --> 00:00:12.440
so I'll try to stay focused.

00:00:12.440 --> 00:00:14.800
So let's think about
whether u-substitution

00:00:14.800 --> 00:00:16.030
might be appropriate.

00:00:16.030 --> 00:00:17.830
Your first temptation
might have said, hey,

00:00:17.830 --> 00:00:22.430
maybe we let u equal sine of 5x.

00:00:22.430 --> 00:00:24.510
And if u is equal
to sine of 5x, we

00:00:24.510 --> 00:00:27.220
have something that's
pretty close to du up here.

00:00:27.220 --> 00:00:28.300
Let's verify that.

00:00:28.300 --> 00:00:32.980
So du could be equal to--
so du dx, derivative of u

00:00:32.980 --> 00:00:33.936
with respect to x.

00:00:33.936 --> 00:00:35.310
Well, we just use
the chain rule.

00:00:35.310 --> 00:00:37.990
Derivative of 5x is 5.

00:00:37.990 --> 00:00:40.900
Times the derivative of sine
of 5x with respect to 5x,

00:00:40.900 --> 00:00:43.700
that's just going
to be cosine of 5x.

00:00:43.700 --> 00:00:45.990
If we want to write this
into differential form, which

00:00:45.990 --> 00:00:48.060
is useful when we do
our u-substitution,

00:00:48.060 --> 00:00:54.500
we could say that du is
equal to 5 cosine 5x.

00:00:54.500 --> 00:00:57.550
And we look over here, we
don't have quite du there.

00:00:57.550 --> 00:01:01.260
We have just cosine of 5x dx.

00:01:01.260 --> 00:01:04.584
Cosine of 5x dx just like that.

00:01:04.584 --> 00:01:06.875
So when you look over here,
you have a cosine of 5x dx,

00:01:06.875 --> 00:01:10.050
but we don't have a
5 cosine of 5x dx.

00:01:10.050 --> 00:01:11.450
But we know how to solve that.

00:01:11.450 --> 00:01:17.380
We can multiply by
5 and divide by 5.

00:01:17.380 --> 00:01:19.800
1/5 times 5 is
just going to be 1,

00:01:19.800 --> 00:01:22.810
so we haven't changed the
value of the expression.

00:01:22.810 --> 00:01:29.390
When we do it this way, we see
pretty clearly we have our u

00:01:29.390 --> 00:01:31.306
and we have our du.

00:01:31.306 --> 00:01:33.100
Our du is 5.

00:01:33.100 --> 00:01:34.000
Let me circle that.

00:01:34.000 --> 00:01:40.650
Let me do that in that blue
color-- is 5 cosine of 5x dx.

00:01:40.650 --> 00:01:43.440
So we can rewrite
this entire expression

00:01:43.440 --> 00:01:47.500
as-- do that 1/5 in purple.

00:01:47.500 --> 00:01:49.130
This is going to
be equal to 1/5--

00:01:49.130 --> 00:01:50.920
I hope you don't hear
that crow outside.

00:01:50.920 --> 00:01:52.360
He's getting quite obnoxious.

00:01:52.360 --> 00:02:03.590
1/5 times the integral of all
the stuff in blue is my du,

00:02:03.590 --> 00:02:08.662
and then that is
over e to the u.

00:02:11.556 --> 00:02:15.420
So how do we take the
antiderivative of this?

00:02:15.420 --> 00:02:20.420
Well, you might be tempted to
well, what would you do here?

00:02:20.420 --> 00:02:23.080
Well, we're still not
quite ready to simply take

00:02:23.080 --> 00:02:24.680
the antiderivative here.

00:02:24.680 --> 00:02:26.830
If I were to rewrite
this, I could rewrite this

00:02:26.830 --> 00:02:33.400
as this is equal to 1/5
times the integral of e

00:02:33.400 --> 00:02:38.810
to the negative u du.

00:02:42.490 --> 00:02:44.170
And so what might
jump out at you

00:02:44.170 --> 00:02:45.950
is maybe we do
another substitution.

00:02:45.950 --> 00:02:49.320
We've already used the letter
u, so maybe we'll use w.

00:02:49.320 --> 00:02:51.150
We'll do some w-substitution.

00:02:51.150 --> 00:02:53.110
And you might be able
to do this in your head,

00:02:53.110 --> 00:02:55.200
but we'll do
w-substitution just to make

00:02:55.200 --> 00:02:57.062
it a little bit clearer.

00:02:57.062 --> 00:02:59.270
This would have been really
useful if this was just e

00:02:59.270 --> 00:03:01.680
to the u because we know the
antiderivative of e to the u

00:03:01.680 --> 00:03:02.430
is just e to the u.

00:03:02.430 --> 00:03:04.429
So let's try to get it
in terms of the form of e

00:03:04.429 --> 00:03:07.240
to the something and not e
to the negative something.

00:03:07.240 --> 00:03:09.210
So let's set.

00:03:09.210 --> 00:03:11.330
And I'm running
out of colors here.

00:03:11.330 --> 00:03:16.280
Let's set w as
equal to negative u.

00:03:16.280 --> 00:03:20.170
In that case, then
dw, derivative

00:03:20.170 --> 00:03:23.052
of w with respect
to u, is negative 1.

00:03:23.052 --> 00:03:26.330
Or if we were to write that
statement in differential form,

00:03:26.330 --> 00:03:33.110
dw is equal to du times
negative 1 is negative du.

00:03:33.110 --> 00:03:36.730
So this right over
here would be our w.

00:03:36.730 --> 00:03:39.110
And do we have a dw here?

00:03:39.110 --> 00:03:40.210
Well, we have just a du.

00:03:40.210 --> 00:03:42.270
We don't have a
negative du there.

00:03:42.270 --> 00:03:46.190
But we can create a negative
du by multiplying this inside

00:03:46.190 --> 00:03:48.780
by a negative 1, but then also
by multiplying the outside

00:03:48.780 --> 00:03:49.535
by a negative 1.

00:03:49.535 --> 00:03:51.460
Negative 1 times
negative 1 is positive 1.

00:03:51.460 --> 00:03:52.860
We haven't changed the value.

00:03:52.860 --> 00:03:55.630
We have to do both of these
in order for it to make sense.

00:03:55.630 --> 00:03:56.930
Or I could do it like this.

00:03:56.930 --> 00:04:01.960
So negative 1 over here and a
negative 1 right over there.

00:04:01.960 --> 00:04:06.990
And if we do it in that form,
then this negative 1 times du,

00:04:06.990 --> 00:04:09.540
that's the same
thing as negative du.

00:04:09.540 --> 00:04:15.020
This is this right over here.

00:04:15.020 --> 00:04:17.380
And so we can
rewrite our integral.

00:04:17.380 --> 00:04:20.459
It's going to be equal to now
it's going to be negative 1/5.

00:04:23.600 --> 00:04:26.100
Trying to use the
colors as best as I can.

00:04:26.100 --> 00:04:31.100
Negative 1/5 times the
indefinite integral of e

00:04:31.100 --> 00:04:33.340
to the-- well,
instead of negative u,

00:04:33.340 --> 00:04:37.670
we can write w-- e to the w.

00:04:37.670 --> 00:04:41.610
And instead of du times
negative 1, or negative du,

00:04:41.610 --> 00:04:42.510
we can write dw.

00:04:46.900 --> 00:04:49.300
Now this simplifies
things a good bit.

00:04:49.300 --> 00:04:52.340
We know what the antiderivative
of this is in terms of w.

00:04:52.340 --> 00:05:02.910
This is going to be equal
to negative 1/5 e to the w.

00:05:06.244 --> 00:05:08.450
And then we might have
some constant there,

00:05:08.450 --> 00:05:11.220
so I'll just do a plus c.

00:05:11.220 --> 00:05:15.120
And now we just have to do
all of our unsubstituting.

00:05:15.120 --> 00:05:19.000
So we know that w is
equal to negative u.

00:05:19.000 --> 00:05:20.020
So we could write that.

00:05:20.020 --> 00:05:22.860
So this is equal
to negative 1/5.

00:05:22.860 --> 00:05:24.740
I want to stay
true to my colors.

00:05:24.740 --> 00:05:35.510
Negative 1/5 e to the negative
u, that's what w is equal to,

00:05:35.510 --> 00:05:36.515
plus c.

00:05:36.515 --> 00:05:38.830
But we're still not
done unsubstituting.

00:05:38.830 --> 00:05:41.842
We know that u is
equal to sine of 5x.

00:05:41.842 --> 00:05:45.520
u is equal to sine of
5x, so we can write this

00:05:45.520 --> 00:05:55.290
as being equal to
negative 1/5 times e

00:05:55.290 --> 00:06:05.100
to the negative u, which is
negative u is sine of 5x.

00:06:05.100 --> 00:06:10.080
And then finally,
we have our plus c.

00:06:10.080 --> 00:06:13.370
Now, there was a simpler way
that we could have done this

00:06:13.370 --> 00:06:14.890
by just doing one substitution.

00:06:14.890 --> 00:06:17.140
But then you would have had
to look ahead a little bit

00:06:17.140 --> 00:06:20.580
and realize that it was not
trivial to take your-- not too

00:06:20.580 --> 00:06:23.871
bad to take your antiderivative
of e to the negative u.

00:06:23.871 --> 00:06:26.370
The insight that you might have
had-- although you shouldn't

00:06:26.370 --> 00:06:28.755
really hold yourself,
feel too bad if you

00:06:28.755 --> 00:06:31.130
didn't see that insight-- is
that we could have rewritten

00:06:31.130 --> 00:06:35.020
that original integral--
and let me rewrite it.

00:06:35.020 --> 00:06:45.280
It's cosine of 5x over
e to the sine of 5x dx.

00:06:45.280 --> 00:06:47.460
We could have written this
entire original integral

00:06:47.460 --> 00:06:52.910
as being equal to
cosine of 5x times e

00:06:52.910 --> 00:06:57.710
to the negative sine of 5x dx.

00:06:57.710 --> 00:07:00.240
And in this situation,
we could have set u

00:07:00.240 --> 00:07:03.710
to be equal to negative
5x and say, well,

00:07:03.710 --> 00:07:06.640
if u is equal to
negative sine of 5x.

00:07:06.640 --> 00:07:10.030
If u is equal to
negative sine of 5x,

00:07:10.030 --> 00:07:16.590
then du is going to be equal
to negative 5 cosine of 5x.

00:07:16.590 --> 00:07:19.282
And we don't have a
negative-- oh, dx.

00:07:19.282 --> 00:07:21.740
We don't have a negative 5
here, but we could construct one

00:07:21.740 --> 00:07:24.440
by putting a negative 5
there and then multiplying

00:07:24.440 --> 00:07:26.780
by negative 1/5.

00:07:26.780 --> 00:07:29.120
And then that would have
immediately simplified

00:07:29.120 --> 00:07:33.110
this integral right over here
to be equal to negative 1/5

00:07:33.110 --> 00:07:40.430
times the integral of
well, we have our du.

00:07:40.430 --> 00:07:43.330
Let me do this in
a different color.

00:07:43.330 --> 00:07:44.640
We have our du.

00:07:44.640 --> 00:07:46.714
That's the negative 5.

00:07:46.714 --> 00:07:47.630
Let me do it this way.

00:07:47.630 --> 00:07:49.855
Negative 5 cosine of 5x dx.

00:07:52.700 --> 00:07:54.030
So that is our du.

00:07:54.030 --> 00:07:56.550
I'm just changing the
order of multiplication.

00:07:56.550 --> 00:08:00.620
Times e to the u.

00:08:00.620 --> 00:08:04.080
This whole thing now is u
this second time around.

00:08:04.080 --> 00:08:06.390
So if we did it this way,
with just one substitution,

00:08:06.390 --> 00:08:09.180
we could have immediately gotten
to the result that we wanted.

00:08:09.180 --> 00:08:11.170
You take the
antiderivative of this.

00:08:11.170 --> 00:08:13.070
I'll do it in one color
now just because I

00:08:13.070 --> 00:08:14.130
think you get the idea.

00:08:14.130 --> 00:08:20.922
This is equal to negative
1/5 e to the u plus c.

00:08:20.922 --> 00:08:24.790
u is equal to
negative sine of 5x.

00:08:24.790 --> 00:08:27.620
So this is equal
to negative 1/5 e

00:08:27.620 --> 00:08:35.020
to the negative sine of
5x plus c And you're done.

00:08:35.020 --> 00:08:36.480
So this one is faster.

00:08:36.480 --> 00:08:37.022
It's simpler.

00:08:37.022 --> 00:08:38.479
And over time, you
might even start

00:08:38.479 --> 00:08:39.950
being able to do
this in your head.

00:08:39.950 --> 00:08:42.419
This top one, you
still didn't mess up

00:08:42.419 --> 00:08:44.409
by just setting u
equal to sine of 5x,

00:08:44.409 --> 00:08:46.450
we just have to do an
extra substitution in order

00:08:46.450 --> 00:08:49.270
to work it through all the way.

00:08:49.270 --> 00:08:53.160
And I was able to do this
video despite the crowing crow

00:08:53.160 --> 00:08:56.460
outside, or squawking crow.