Solid of Revolution (part 6)

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:05.310
In the last video we took
essentially the length of y

00:00:05.310 --> 00:00:09.620
equals x square, not the
length, but we went from zero

00:00:09.620 --> 00:00:14.340
to 1 on the x-axis and you
can view it as this area.

00:00:14.340 --> 00:00:17.710
And we rotated around the
y-axis to get this figure here

00:00:17.710 --> 00:00:19.470
and we figured out the volume,
I think our answer if I

00:00:19.470 --> 00:00:21.330
remember was pi over 2.

00:00:21.330 --> 00:00:24.160
And we use what I called
and what everyone calls

00:00:24.160 --> 00:00:25.470
the Shell method.

00:00:25.470 --> 00:00:27.090
I want to show you that
you could actually used

00:00:27.090 --> 00:00:28.740
the disk method here.

00:00:28.740 --> 00:00:30.880
But then we'll just have
to switch the ys and xs.

00:00:30.880 --> 00:00:36.230
Instead of writing this the
function as a function of x,

00:00:36.230 --> 00:00:38.500
we'll just take the
inverse of it and write it

00:00:38.500 --> 00:00:39.320
as a function of y.

00:00:39.320 --> 00:00:42.270
so this curve y equals x
squared, what it can also be

00:00:42.270 --> 00:00:43.900
written as, we can just take
the squared root of both

00:00:43.900 --> 00:00:48.620
sides as x is equal to
the square root of y.

00:00:48.620 --> 00:00:50.630
Just take the square
root of both sides.

00:00:50.630 --> 00:00:53.440
And now we can use this
information to do the disk

00:00:53.440 --> 00:00:56.650
method, but everywhere where
we had an x in the past

00:00:56.650 --> 00:00:57.500
we'll now have a y.

00:00:57.500 --> 00:00:59.560
So let's think about
how we would do it.

00:00:59.560 --> 00:01:04.240
Once again it's this area--
really the hardest thing about

00:01:04.240 --> 00:01:07.110
all of these problems is
just the visualization.

00:01:07.110 --> 00:01:08.040
I think that's why they do it.

00:01:08.040 --> 00:01:10.230
Just to make sure that you know
how to visualize things and

00:01:10.230 --> 00:01:11.560
maybe understand the calculus.

00:01:11.560 --> 00:01:14.490
It's more visualization
then calculus really.

00:01:14.490 --> 00:01:18.060
So we're still dealing with--
if we take a cross section, if

00:01:18.060 --> 00:01:23.270
we were to cut this figure this
area would be the

00:01:23.270 --> 00:01:23.885
cross section.

00:01:26.570 --> 00:01:28.930
And we're still rotating around
the y-axis just like we

00:01:28.930 --> 00:01:29.900
did in the last video.

00:01:29.900 --> 00:01:32.290
We're still rotation around the
y-axis, we get this figure.

00:01:32.290 --> 00:01:34.420
So how would we
deal with disks.

00:01:34.420 --> 00:01:39.266
Well a disk would look like
this at some point-- let

00:01:39.266 --> 00:01:41.690
me pick another color.

00:01:41.690 --> 00:01:46.540
At some point we'd have
a disk like that.

00:01:46.540 --> 00:01:50.330
That would be the top of the
disk, and it would have some

00:01:50.330 --> 00:01:53.520
depth like we did before.

00:01:58.070 --> 00:02:02.310
And its height, or the
radius of that disk,

00:02:02.310 --> 00:02:06.220
would be equal to x.

00:02:06.220 --> 00:02:07.230
It's equal to x.

00:02:07.230 --> 00:02:08.640
I know you're thinking it
looks like the shell method.

00:02:08.640 --> 00:02:09.980
But what is x equal to?

00:02:09.980 --> 00:02:12.130
It's equal to the
square root of y.

00:02:12.130 --> 00:02:14.160
It equals the square root of y.

00:02:14.160 --> 00:02:16.810
And what would be the
width of that disk?

00:02:16.810 --> 00:02:19.400
Now we are making everything as
a function of y, so the width

00:02:19.400 --> 00:02:22.430
would be just a very small
distance d y, the

00:02:22.430 --> 00:02:23.420
differential y.

00:02:28.030 --> 00:02:29.500
That's essentially
all we need to know.

00:02:29.500 --> 00:02:32.870
So the volume of that disk
would be the radius squared

00:02:32.870 --> 00:02:36.670
times pi times d y.

00:02:36.670 --> 00:02:40.070
Hopefully that makes a little
bit sense, but there's another

00:02:40.070 --> 00:02:41.070
hitch on this problem.

00:02:41.070 --> 00:02:43.200
This is actually similar to
what we did two videos ago.

00:02:43.200 --> 00:02:48.750
Because when you view it in the
y-axis, from this y frame of

00:02:48.750 --> 00:02:51.440
reference, what we're going to
do is we're going to take the

00:02:51.440 --> 00:02:57.660
volume, we take the volume
of x is equal to 1.

00:02:57.660 --> 00:03:00.190
So what would be the volume
of x is equal to 1 rotated

00:03:00.190 --> 00:03:01.100
around the x-axis?

00:03:01.100 --> 00:03:03.933
It would just be the
entire cylinder.

00:03:06.930 --> 00:03:08.410
It's really important
that you visualize this

00:03:08.410 --> 00:03:10.780
right, what we're doing.

00:03:10.780 --> 00:03:15.400
We're going to figure
out this volume, volume

00:03:15.400 --> 00:03:16.740
of x is equal to 1.

00:03:21.030 --> 00:03:23.990
Let me draw the axis just so
you know what we're doing.

00:03:23.990 --> 00:03:30.240
So this would be the y-axis,
that would be the x-axis

00:03:30.240 --> 00:03:31.220
as best as I can draw.

00:03:34.010 --> 00:03:35.660
Originally, we can figure
out the volume of

00:03:35.660 --> 00:03:37.240
the entire cylinder.

00:03:37.240 --> 00:03:43.740
This is x is equal to 1, x is
equal to 1 from what y points?

00:03:43.740 --> 00:03:44.960
From the point,
well what is this?

00:03:44.960 --> 00:03:47.760
This is y equals 1.

00:03:47.760 --> 00:03:49.930
y equal 1 to zero.

00:03:49.930 --> 00:03:54.230
So we would figure out this
volume from y equals 1 to zero.

00:03:54.230 --> 00:03:56.000
And how would we do that?

00:03:56.000 --> 00:03:58.750
What would be the integral?

00:03:58.750 --> 00:04:01.390
Remember everything is in
y now, so it might seem

00:04:01.390 --> 00:04:03.660
a little confusing.

00:04:03.660 --> 00:04:07.180
For each of these disks, this
is going to be made up of

00:04:07.180 --> 00:04:09.860
a bunch of disks, let
me draw one of them.

00:04:09.860 --> 00:04:12.060
Let me draw the top one.

00:04:12.060 --> 00:04:16.170
The top disk, and it has a
width, it's not going to be

00:04:16.170 --> 00:04:18.600
this entire cylinder, its
width is d y, its width

00:04:18.600 --> 00:04:21.410
is just going to be this.

00:04:21.410 --> 00:04:24.560
d y, a very small sliver.

00:04:24.560 --> 00:04:32.160
d y, and its radius is x
or you could say f of y.

00:04:35.660 --> 00:04:38.040
What would be the
volume of that disk?

00:04:38.040 --> 00:04:40.290
What would be the surface
area of the top?

00:04:40.290 --> 00:04:44.350
It would be f of y squared.

00:04:44.350 --> 00:04:46.740
Remember we're dealing
everything with y. f of y

00:04:46.740 --> 00:04:50.410
squared times pi, that
gives the area of the top.

00:04:50.410 --> 00:04:54.510
Put the pi outside of
the integral times d y.

00:04:54.510 --> 00:04:58.700
d y, and we said y is
going from zero to 1.

00:04:58.700 --> 00:05:00.800
y is going from zero to 1.

00:05:00.800 --> 00:05:03.130
So that's that entire cylinder.

00:05:03.130 --> 00:05:05.860
And what we'll want to do is
subtract out, essentially cut

00:05:05.860 --> 00:05:09.010
out the volume of
the inner bowl.

00:05:09.010 --> 00:05:10.730
So minus.

00:05:10.730 --> 00:05:12.580
How would we figure
out the inner bowl?

00:05:12.580 --> 00:05:16.410
That's where we will deal
with the x is equal to the

00:05:16.410 --> 00:05:19.240
square root of y function.

00:05:19.240 --> 00:05:20.650
Because here, what's each disk?

00:05:20.650 --> 00:05:22.830
Once again it's f of y.

00:05:22.830 --> 00:05:24.980
I wrote this generally, this
is f of y of the outside.

00:05:27.580 --> 00:05:29.890
I'm going to now do
it for the inside.

00:05:29.890 --> 00:05:32.980
I'm going to cut out the inside
volume and because I kept

00:05:32.980 --> 00:05:34.160
it general I could do it.

00:05:34.160 --> 00:05:36.360
We're still going from
zero to 1, from y equals

00:05:36.360 --> 00:05:37.640
zero to y equals 1.

00:05:37.640 --> 00:05:40.070
Remember we're
doing with y now.

00:05:40.070 --> 00:05:47.490
I will take f of the inside
f of y squared d of y.

00:05:47.490 --> 00:05:50.720
You can imagine in this case
the inside function, we're

00:05:50.720 --> 00:05:51.850
going to take the volume.

00:05:51.850 --> 00:05:53.890
That original disk I drew is
actually one of the disks

00:05:53.890 --> 00:05:55.260
for the inside function.

00:05:55.260 --> 00:06:00.640
Where the height of that disk
is d y, that should be d y.

00:06:00.640 --> 00:06:03.050
The radius of the
disk is f of y.

00:06:03.050 --> 00:06:06.260
And of course the area of
the top of the desk is pi

00:06:06.260 --> 00:06:08.160
times the radius squared.

00:06:08.160 --> 00:06:10.840
How do we apply it to this
particular situation?

00:06:10.840 --> 00:06:12.140
What is the outside function?

00:06:12.140 --> 00:06:15.790
f of y, x is equal to f of y.

00:06:15.790 --> 00:06:17.370
We are just switching
the variables.

00:06:17.370 --> 00:06:18.840
We say that x is equal to 1.

00:06:18.840 --> 00:06:20.590
This is just a big
cylinder, right?

00:06:20.590 --> 00:06:23.306
So f of i is equal go to 1 in
this function, so we get pi--

00:06:23.306 --> 00:06:28.420
let me switch colors--
times from zero to 1.

00:06:28.420 --> 00:06:35.530
1 squared d y minus pi,
and we're still going

00:06:35.530 --> 00:06:38.170
from zero to 1.

00:06:38.170 --> 00:06:41.050
Remember, that's
the y boundaries.

00:06:41.050 --> 00:06:42.570
What's f of y?

00:06:42.570 --> 00:06:51.540
Here f of y is square
root of y squared d y.

00:06:55.330 --> 00:06:56.440
Let's take the pi out.

00:07:00.010 --> 00:07:04.280
So pi times, well
1 squared is 1.

00:07:04.280 --> 00:07:06.060
What's the
anti-derivative of 1?

00:07:06.060 --> 00:07:08.580
What function's
derivative is 1?

00:07:08.580 --> 00:07:11.371
It's x, right?

00:07:11.371 --> 00:07:13.110
x, that's the
anti-derivative there.

00:07:13.110 --> 00:07:14.600
We can merge these.

00:07:14.600 --> 00:07:16.120
We took the pi out.

00:07:16.120 --> 00:07:20.210
This square root of y
squared, this is just y.

00:07:20.210 --> 00:07:21.880
Sorry, we have that
derivative 1.

00:07:21.880 --> 00:07:23.780
See, even I get quite
confused here.

00:07:23.780 --> 00:07:26.550
We're taking this with
respect to y, right?

00:07:26.550 --> 00:07:29.200
The anti-derivative
of 1 now is y.

00:07:29.200 --> 00:07:31.930
We are setting up
a bunch of ys.

00:07:31.930 --> 00:07:32.960
I'm sorry.

00:07:32.960 --> 00:07:34.700
I find this even a little
perplexing when I start

00:07:34.700 --> 00:07:35.580
switching x and y.

00:07:38.350 --> 00:07:39.980
The second function.

00:07:39.980 --> 00:07:41.540
Remember, we can merge these
because it's the same

00:07:41.540 --> 00:07:43.520
boundaries and they're
both in respect to y.

00:07:43.520 --> 00:07:45.330
Square root of y squared is y.

00:07:45.330 --> 00:07:49.940
What's the anti-derivative of
y, the function y or y d y.

00:07:49.940 --> 00:07:57.320
Was y squared over 2,
minus y squared over 2.

00:07:57.320 --> 00:08:00.490
We are going to evaluate
that at 1 and zero.

00:08:03.160 --> 00:08:04.000
So what does that get us?

00:08:04.000 --> 00:08:12.680
We get pi-- we get 1 minus
1/2 minus zero minus zero

00:08:12.680 --> 00:08:15.290
minus zero plus 0/2.

00:08:15.290 --> 00:08:16.600
Whatever these are zero.

00:08:16.600 --> 00:08:21.870
1 minus 1/2 that's
1/2 and times pi is

00:08:21.870 --> 00:08:23.540
equal to pie over 2.

00:08:23.540 --> 00:08:24.930
Which is the exact same result.

00:08:24.930 --> 00:08:28.010
And I was worried, because you
never know when you might make

00:08:28.010 --> 00:08:29.940
a careless mistake
as I often do.

00:08:29.940 --> 00:08:32.650
We got the exact same result as
I got in the previous video

00:08:32.650 --> 00:08:34.850
when we used the shell method.

00:08:34.850 --> 00:08:37.050
The hardest thing here is just
remembering that you're doing

00:08:37.050 --> 00:08:40.080
everything in terms of y.

00:08:40.080 --> 00:08:41.710
When we did the disk method
traditionally, the disks

00:08:41.710 --> 00:08:43.050
were vertical disks.

00:08:43.050 --> 00:08:45.930
Now they're horizontal disks
but it's the exact same thing.

00:08:45.930 --> 00:08:48.900
You just have to get your brain
around the idea that we're

00:08:48.900 --> 00:08:51.780
dealing with the ys, that the
boundaries on the integrals

00:08:51.780 --> 00:08:53.360
are now y values.

00:08:53.360 --> 00:08:58.630
We're taking the width of the
disks, or the height of the

00:08:58.630 --> 00:09:01.890
disks are d y and the radius
is now the function of y.

00:09:01.890 --> 00:09:05.730
Hopefully I didn't
confuse you too much.

00:09:05.730 --> 00:09:07.890
My own brain malfunctioned
a bit when I took the

00:09:07.890 --> 00:09:11.740
anti-derivative of 1 d y,
it should have been y.

00:09:11.740 --> 00:09:13.900
I will see you in the next
video where we will do even

00:09:13.900 --> 00:09:16.290
more complicated problems,
where I'm sure I'll make

00:09:16.290 --> 00:09:16.970
even more mistakes.

00:09:16.970 --> 00:09:18.710
See you soon.


Solid of Revolution (part 5)

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.550
We've been doing a lot of
rotating around the x-axis, so

00:00:03.550 --> 00:00:07.870
let's start rotating around the
y-axis and see what we can do.

00:00:07.870 --> 00:00:10.620
Or at least attempt to.

00:00:10.620 --> 00:00:12.150
Let's me draw my axes.

00:00:12.150 --> 00:00:14.740
That's y-axis.

00:00:14.740 --> 00:00:15.770
That's my x-axis.

00:00:18.470 --> 00:00:21.173
Well let's just do it with an
example, but we'll call it f

00:00:21.173 --> 00:00:24.460
of x too because it'll
be generalizable.

00:00:24.460 --> 00:00:26.635
Let's just draw y
equals x squared.

00:00:30.420 --> 00:00:33.450
Let me just draw the positive
because we're going to rotate

00:00:33.450 --> 00:00:35.900
it around the y-axis and it's
symmetric anyway, so that's

00:00:35.900 --> 00:00:37.160
y equals x squared.

00:00:37.160 --> 00:00:39.650
This is y-axis.

00:00:39.650 --> 00:00:41.870
This is x-axis.

00:00:41.870 --> 00:00:43.840
Actually, no I'm going to keep
it general, then we'll actually

00:00:43.840 --> 00:00:44.660
solve it particularly.

00:00:44.660 --> 00:00:47.820
So we'll call this f of
x, but clearly this is

00:00:47.820 --> 00:00:48.990
y equals x squared.

00:00:48.990 --> 00:00:50.900
This is f of x.

00:00:50.900 --> 00:00:53.830
And we know how to take the
volume if I were to rotate

00:00:53.830 --> 00:00:54.770
this around the x-axis.

00:00:54.770 --> 00:01:00.720
But what if I wanted to say-- I
guess we could call it the area

00:01:00.720 --> 00:01:08.030
between 0 and-- I'm trying to
determine how general to be.

00:01:08.030 --> 00:01:09.400
Well let's just say
between 0 and 1.

00:01:09.400 --> 00:01:13.120
I think the boundaries
might make sense to you.

00:01:13.120 --> 00:01:17.360
Roughly this area, and
I'm going to rotate it

00:01:17.360 --> 00:01:21.240
around the y-axis now.

00:01:21.240 --> 00:01:25.460
So what's that final figure
going to look like?

00:01:25.460 --> 00:01:28.870
The base of it-- let me see
how well I can draw it.

00:01:28.870 --> 00:01:32.390
Nope that's not what
I wanted to do.

00:01:32.390 --> 00:01:33.790
The base is going to
look something like a

00:01:33.790 --> 00:01:35.750
cylinder like that.

00:01:35.750 --> 00:01:40.830
And then the top of it is also
going to be-- no, that's

00:01:40.830 --> 00:01:42.620
not what I wanted to do.

00:01:42.620 --> 00:01:44.560
Let me draw the side lines.

00:01:44.560 --> 00:01:49.110
So it's going to look
something like that.

00:01:55.170 --> 00:02:03.790
And then the top of it
looks something like that.

00:02:03.790 --> 00:02:05.440
But it's not just going
to be cylinder, right?

00:02:05.440 --> 00:02:07.200
If I was doing this entire
block it would be a cylinder.

00:02:07.200 --> 00:02:09.630
But the inside of it is going
to be kind of hollowed out.

00:02:09.630 --> 00:02:12.530
Let me see how effective
I am at drawing that.

00:02:12.530 --> 00:02:14.340
I'll do it in a
different color.

00:02:14.340 --> 00:02:15.790
So the inside is going
to be hollowed out.

00:02:19.510 --> 00:02:21.350
I don't know if that
makes sense to you that.

00:02:21.350 --> 00:02:25.540
It's kind of like on the inside
it'll look like a bowl.

00:02:25.540 --> 00:02:29.310
On the outside it'll look
like a cylinder or a can.

00:02:29.310 --> 00:02:29.990
Hopefully that makes sense.

00:02:29.990 --> 00:02:31.800
You take this and you
rotate this around.

00:02:31.800 --> 00:02:36.660
And the curve that specifies
the inside would be y

00:02:36.660 --> 00:02:40.060
is equal to x squared.

00:02:40.060 --> 00:02:42.890
It would rotate all
the way around.

00:02:42.890 --> 00:02:43.730
I think that makes sense.

00:02:43.730 --> 00:02:45.340
The drawing is the
hardest part.

00:02:45.340 --> 00:02:46.780
So how do we do it?

00:02:46.780 --> 00:02:48.900
Well even the shape
might give you an idea.

00:02:48.900 --> 00:02:51.910
We can't use that disk method,
what we were doing before when

00:02:51.910 --> 00:02:54.090
we were rotating the x-axis,
that was the disk method,

00:02:54.090 --> 00:02:56.600
because we were essentially
imagining each of these

00:02:56.600 --> 00:02:59.140
particular disks and
then summing them up.

00:02:59.140 --> 00:03:01.510
Now we're going to do something
called the shell method.

00:03:01.510 --> 00:03:04.460
So what's the shell method?

00:03:04.460 --> 00:03:08.460
Instead of taking a bunch of
disks and figuring out their

00:03:08.460 --> 00:03:11.180
combined volumes, we're going
to take a bunch of shells.

00:03:11.180 --> 00:03:12.430
So what's a shell?

00:03:12.430 --> 00:03:16.570
So imagine a rectangle
right here.

00:03:16.570 --> 00:03:20.280
Hope you can see
it right there.

00:03:20.280 --> 00:03:24.710
Let's say it's at
the point x,1.

00:03:24.710 --> 00:03:26.300
What's its height going to be?

00:03:26.300 --> 00:03:27.645
Its height going
to be f of x,1.

00:03:31.200 --> 00:03:32.990
That's its height.

00:03:32.990 --> 00:03:37.150
Now imagine taking that
sliver and rotating

00:03:37.150 --> 00:03:39.810
it around the y-axis.

00:03:39.810 --> 00:03:40.650
What's it going to look like?

00:03:40.650 --> 00:03:43.550
Well, it's going to look like a
shell, it's going to look like

00:03:43.550 --> 00:03:47.910
a cylinder, just like the
outside of a cylinder.

00:03:47.910 --> 00:03:50.120
It's going to look not too
different then that but I want

00:03:50.120 --> 00:03:53.950
to draw it well because
intuition is the most important

00:03:53.950 --> 00:03:57.720
thing that, not getting
the problem right.

00:03:57.720 --> 00:04:01.810
Let me see if I can
draw this respectably.

00:04:01.810 --> 00:04:05.230
And then we're going to have
the bottom of the shell, it'll

00:04:05.230 --> 00:04:08.520
look something like that.

00:04:08.520 --> 00:04:10.910
Let me finish these lines up.

00:04:10.910 --> 00:04:13.400
I think you get the point.

00:04:13.400 --> 00:04:14.000
OK.

00:04:14.000 --> 00:04:15.700
So it's going to look
like a shell like that.

00:04:22.770 --> 00:04:25.385
The outside of the shell
is going to be solid.

00:04:29.700 --> 00:04:34.450
And it'll have some width,
but the inside is hollow.

00:04:34.450 --> 00:04:36.390
Let me do a different color.

00:04:36.390 --> 00:04:38.395
Maybe a darker color to show
that that's the inside.

00:04:42.370 --> 00:04:44.610
You know it's like a
ring essentially.

00:04:44.610 --> 00:04:46.980
And so what's the
height of this ring?

00:04:46.980 --> 00:04:50.310
The height is going
to be f of x,1.

00:04:50.310 --> 00:04:53.070
So let me do a brighter color
so you know what I'm saying.

00:04:53.070 --> 00:04:57.470
The height of this
ring is f of x,1.

00:04:57.470 --> 00:05:01.530
f of x evaluated at that
arbitrary point we picked up.

00:05:01.530 --> 00:05:06.430
What is going to be the
surface area of this ring?

00:05:06.430 --> 00:05:08.300
You know, this outside.

00:05:08.300 --> 00:05:08.970
Well let's think about it.

00:05:08.970 --> 00:05:12.880
It'll be the circumference of
this ring times it's height.

00:05:12.880 --> 00:05:14.530
So what's the circumference
of this ring?

00:05:18.050 --> 00:05:19.560
Let's go back to our
basic geometry.

00:05:19.560 --> 00:05:24.280
Circumference is equal to
2 pi times the radius.

00:05:24.280 --> 00:05:28.790
So if we know the radius of it,
we know the circumference.

00:05:28.790 --> 00:05:29.790
Well what's the radius?

00:05:29.790 --> 00:05:32.750
Well the radius is how far
we went from the axis of

00:05:32.750 --> 00:05:35.510
rotation to that point.

00:05:35.510 --> 00:05:36.395
So that's the radius.

00:05:39.460 --> 00:05:41.880
So in our particular
example the radius is x,1.

00:05:44.480 --> 00:05:47.610
It's that x point that
we're evaluating it at.

00:05:47.610 --> 00:05:52.330
So circumference is going to be
equal to 2 pi times that point

00:05:52.330 --> 00:05:53.680
that we're evaluating at.

00:05:53.680 --> 00:05:57.640
And so the surface area-- this
magenta thing that I filled

00:05:57.640 --> 00:06:02.060
in-- that's going to be equal
to the circumference times this

00:06:02.060 --> 00:06:03.740
height, which we already
said is f of x,1.

00:06:07.040 --> 00:06:09.900
Let's call it area surface.

00:06:09.900 --> 00:06:13.610
Surface area is equal to
circumference times height,

00:06:13.610 --> 00:06:21.830
which is equal to 2 pi
x,1 times f of x,1.

00:06:21.830 --> 00:06:24.640
We figured out the
surface are of this.

00:06:24.640 --> 00:06:25.705
Now how do we figure
out the volume?

00:06:28.560 --> 00:06:29.840
Well what's the width of it?

00:06:29.840 --> 00:06:32.040
How thick is this ring?

00:06:32.040 --> 00:06:33.630
What's this thickness
right here?

00:06:33.630 --> 00:06:35.005
It's a very small thickness.

00:06:35.005 --> 00:06:38.250
But we took this sliver, and
this sliver as we learned in

00:06:38.250 --> 00:06:42.770
previous calculus, the width of
this little rectangle is dx.

00:06:42.770 --> 00:06:44.270
And you know when we take the
integral, it's going to get

00:06:44.270 --> 00:06:46.670
infinitely smaller and smaller
and we'll have infinitely

00:06:46.670 --> 00:06:47.740
more and more of them.

00:06:47.740 --> 00:06:49.870
So the width of this is dx.

00:06:49.870 --> 00:06:53.990
Let me draw it big, not
so horrible looking.

00:06:53.990 --> 00:06:59.880
So if this is a sliver,
it's width is dx.

00:06:59.880 --> 00:07:02.980
It's height is f of x,1.

00:07:02.980 --> 00:07:04.840
x,1 will be right
in the center.

00:07:04.840 --> 00:07:08.600
And then it's distance from
the center is of course x,1.

00:07:08.600 --> 00:07:09.980
Hopefully that make sense.

00:07:09.980 --> 00:07:13.260
So what's the volume
of this shell?

00:07:13.260 --> 00:07:16.450
So the volume of the shell--
this shell, not this one-- the

00:07:16.450 --> 00:07:19.570
volume of the shell is going to
be equal to the surface area

00:07:19.570 --> 00:07:23.020
of the shell times how
wide that surface is.

00:07:23.020 --> 00:07:26.490
And that width is dx, so it's
going to equal this times dx.

00:07:26.490 --> 00:07:31.480
So the volume of that
shell is 2 pi x,1 times

00:07:31.480 --> 00:07:36.350
f of x,1 times dx.

00:07:36.350 --> 00:07:39.350
I think you see where I'm
going with this now.

00:07:39.350 --> 00:07:43.170
So what would be the volume
of the entire rotated

00:07:43.170 --> 00:07:44.400
figure, this thing here?

00:07:44.400 --> 00:07:47.060
Well I'm just going to sum
up each of these shells.

00:07:47.060 --> 00:07:49.920
I have one shell there, then
here I'll have a slightly less

00:07:49.920 --> 00:07:52.880
high shell, and up here I would
have a much bigger shell,

00:07:52.880 --> 00:07:54.040
and I'll add them up.

00:07:54.040 --> 00:07:57.100
Here's one shell
that goes around.

00:07:57.100 --> 00:07:59.200
Then they'll be another shell
here, and I'll add them all up.

00:07:59.200 --> 00:08:00.810
And that's taking the integral.

00:08:00.810 --> 00:08:03.820
So the total volume of the
figure when I rotate it around

00:08:03.820 --> 00:08:12.020
the y-axis is going to be-- and
my boundary is from 0 to 1-- 2

00:08:12.020 --> 00:08:17.110
pi-- this one I just told you a
particular x,1 but we're going

00:08:17.110 --> 00:08:18.560
to sum them over
all of the x's.

00:08:18.560 --> 00:08:25.550
So it's going to be
2 pi x f of x dx.

00:08:25.550 --> 00:08:27.160
This is just a constant,
so you could call it

00:08:27.160 --> 00:08:30.110
2 pi times x f of x.

00:08:30.110 --> 00:08:31.550
So let's take a
particular example.

00:08:31.550 --> 00:08:33.290
Let's do it for x squared.

00:08:33.290 --> 00:08:35.370
Let's say the function
is x squared.

00:08:35.370 --> 00:08:37.786
So in this case the volume is
going to equal-- let's take the

00:08:37.786 --> 00:08:44.500
2 pi out-- 2 pi integral 0 to 1
x times f of x-- f of x in our

00:08:44.500 --> 00:08:50.650
case is x squared, which I drew
earlier-- dx equals 2 pi.

00:08:50.650 --> 00:08:52.840
This is just x to
the third, right?

00:08:52.840 --> 00:08:53.720
x to the third.

00:08:53.720 --> 00:08:56.080
So it's going to be 2 pi
times the antiderivative

00:08:56.080 --> 00:08:57.290
of x to the third.

00:08:57.290 --> 00:08:58.940
Well that's x to
the fourth over 4.

00:09:01.490 --> 00:09:05.280
Evaluate it at 1 minus
evaluate it at 0.

00:09:05.280 --> 00:09:11.630
Well that equals 2 pi times
1 to the fourth is 1, so

00:09:11.630 --> 00:09:16.220
1/4 and then minus 0.

00:09:16.220 --> 00:09:18.690
So it's 2 pi times 1/4.

00:09:18.690 --> 00:09:20.140
So that's pi over 2.

00:09:22.770 --> 00:09:25.990
That's the volume, and we just
rotated it around the y-axis.

00:09:25.990 --> 00:09:27.360
I will see you in
the next video.


Solid of Revolution (part 4)

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:01.060
Let's

00:00:01.060 --> 00:00:06.420
continue on with our study of
rotation of functions around

00:00:06.420 --> 00:00:09.375
the x, and we'll soon
see the y-axis as well.

00:00:09.375 --> 00:00:12.040
So let's do a slightly harder
example than what we've been

00:00:12.040 --> 00:00:15.360
doing, but I think it might be
obvious how to approach it.

00:00:15.360 --> 00:00:21.300
So there's my y-axis, there's
my x-axis, and in a couple of--

00:00:21.300 --> 00:00:24.290
I think it was two problems
ago-- we figured out if we had

00:00:24.290 --> 00:00:27.530
the function y is equal to
square root of x-- let me try

00:00:27.530 --> 00:00:33.210
to draw it-- so this is y
equals square root of x-- if we

00:00:33.210 --> 00:00:37.460
were rotate that around the
x-axis, what the volume would

00:00:37.460 --> 00:00:41.410
be between two points, let's
say 0 and some other point.

00:00:41.410 --> 00:00:42.940
Now let's just pick an
arbitrary point 1.

00:00:42.940 --> 00:00:45.390
I think you know how to
do that at this point.

00:00:45.390 --> 00:00:48.320
Now let's make it a slightly
more difficult problem.

00:00:48.320 --> 00:00:51.785
Let's say I were to also
draw the function y is

00:00:51.785 --> 00:00:52.760
equal to x squared.

00:00:52.760 --> 00:00:56.115
So that looks-- if this is 1,
they both meet at 1, right?

00:00:56.115 --> 00:01:00.900
Because square root of 1 is 1,
and 1 squared is 1, so that

00:01:00.900 --> 00:01:02.380
would look something like this.

00:01:07.150 --> 00:01:08.750
They should be actually
symetric around y

00:01:08.750 --> 00:01:11.650
equals x, but anyway.

00:01:11.650 --> 00:01:16.010
So say y equals x squared
looks like that.

00:01:16.010 --> 00:01:19.940
So my question now is, what
is the volume if I were to

00:01:19.940 --> 00:01:22.490
take this figure and
rotate it around?

00:01:22.490 --> 00:01:23.530
So what do I mean?

00:01:23.530 --> 00:01:33.940
So this area here, if I were to
rotate that about the x-axis,

00:01:33.940 --> 00:01:35.150
what would the volume be?

00:01:35.150 --> 00:01:38.350
So now, what we did just with
the square root of x, we had

00:01:38.350 --> 00:01:39.730
like a solid cup, right?

00:01:39.730 --> 00:01:40.840
It would look
something like this.

00:01:40.840 --> 00:01:44.200
It would be like a cup, and it
was solid, we were just trying

00:01:44.200 --> 00:01:45.500
to figure out the volume of it.

00:01:45.500 --> 00:01:47.730
Hopefully that made
sense to you.

00:01:47.730 --> 00:01:51.610
Now it's going to be kind of a
hollowed-out cup, because we

00:01:51.610 --> 00:01:54.400
have this inside function, and
so the inside of the cup

00:01:54.400 --> 00:01:56.940
is going to be empty.

00:01:56.940 --> 00:01:57.910
Hopefully that makes
a little sense.

00:01:57.910 --> 00:02:00.250
Remember you're just taking
this and then you're rotating

00:02:00.250 --> 00:02:03.290
it around the x-axis.

00:02:03.290 --> 00:02:05.390
Well, the way to think about
it-- especially if you having

00:02:05.390 --> 00:02:07.590
trouble visualizing--
actually, the solution might

00:02:07.590 --> 00:02:09.040
help you visualize it.

00:02:09.040 --> 00:02:11.640
The volume of this figure,
which I'm having trouble

00:02:11.640 --> 00:02:15.180
drawing, it will be the
volume formed by the

00:02:15.180 --> 00:02:19.885
outside rotation of this y
equals square root of x.

00:02:19.885 --> 00:02:20.980
We'll do that in the yellow.

00:02:24.630 --> 00:02:27.740
It'll be the volume formed when
that is rotated around, and

00:02:27.740 --> 00:02:31.770
then the whole solid volume
minus the volume when

00:02:31.770 --> 00:02:35.950
minus this volume.

00:02:35.950 --> 00:02:43.420
So if we took the y equals x
squared, y equals x squared

00:02:43.420 --> 00:02:46.060
would look something like that,
and then if you rotated it

00:02:46.060 --> 00:02:50.780
around the axis, it would
look something like that.

00:02:50.780 --> 00:02:52.870
I don't know if you've ever
been to Morocco, but they have

00:02:52.870 --> 00:02:56.020
these tajin plates that the
tops look a lot like that.

00:02:56.020 --> 00:02:56.730
Well, you probably haven't.

00:02:56.730 --> 00:02:57.920
Well, anyway.

00:02:57.920 --> 00:02:59.230
It would look
something like that.

00:02:59.230 --> 00:03:05.200
So if we subtract out this
volume when it's rotated around

00:03:05.200 --> 00:03:08.480
from the volume of y equals
square root of x, when that's

00:03:08.480 --> 00:03:10.340
rotated, we'll get this figure
that we're trying to figure

00:03:10.340 --> 00:03:12.230
out, this area when
it's rotated around.

00:03:12.230 --> 00:03:15.070
And that should be intuitive
for you, hopefully, because

00:03:15.070 --> 00:03:17.260
when we just did area under a
curve, that's how we would

00:03:17.260 --> 00:03:18.660
figure out the area
of this green area.

00:03:18.660 --> 00:03:21.140
We would figure out the area
under square root of x, and

00:03:21.140 --> 00:03:24.900
we'd subtract out the area
under y equals x squared.

00:03:28.400 --> 00:03:32.690
This time, we're going to say
the volume of the revolution of

00:03:32.690 --> 00:03:35.950
y equals square root of x minus
the volume of the revolution

00:03:35.950 --> 00:03:37.340
of y equals x squared.

00:03:37.340 --> 00:03:41.260
So let's do the problem.

00:03:41.260 --> 00:03:52.280
So the total volume-- let me do
a good color, that looks good--

00:03:52.280 --> 00:03:56.250
total volume is going to be
equal to the volume formed when

00:03:56.250 --> 00:04:01.810
we rotate y equals square
root of x around the x-axis.

00:04:01.810 --> 00:04:04.010
I said from 0 to 1, and
that's because I picked

00:04:04.010 --> 00:04:05.440
where they meet.

00:04:05.440 --> 00:04:08.570
Sometimes in a book, or on an
exam, they'll just say, oh, you

00:04:08.570 --> 00:04:12.350
know, the area between y equals
x squared and y equals square

00:04:12.350 --> 00:04:13.980
of x, we're going to rotate
that around and you have

00:04:13.980 --> 00:04:16.880
to figure out, well they
intersected at 1, and you can

00:04:16.880 --> 00:04:19.270
just set the equations equal to
each other to figure that out.

00:04:19.270 --> 00:04:20.960
We're going from 0 to
1, because they also

00:04:20.960 --> 00:04:22.550
intersect at 0.

00:04:22.550 --> 00:04:24.910
0 squared is the same thing
as square root of 0.

00:04:24.910 --> 00:04:27.150
We're going from 0 to
1, and so what's the

00:04:27.150 --> 00:04:30.280
volume of the larger?

00:04:30.280 --> 00:04:33.720
Or I guess the y equals square
root of x rotated around?

00:04:33.720 --> 00:04:36.800
I always forget the formula,
that's why I always redraw a

00:04:36.800 --> 00:04:43.660
disk, so if that's the radius
of my disk, the disk is going

00:04:43.660 --> 00:04:46.700
to come around like that, so
we know that the radius is a

00:04:46.700 --> 00:04:50.910
function of the disk, and
that's, of course, the dx

00:04:50.910 --> 00:04:54.160
is the depth of the disk.

00:04:54.160 --> 00:04:58.290
So the radius is the function
which, for the outside one, is

00:04:58.290 --> 00:05:04.360
square root of x, and we know
area of this disk is pi r

00:05:04.360 --> 00:05:11.010
squared, so we square the
radius, take a pi outside, and

00:05:11.010 --> 00:05:13.990
then we multiply that times the
width, so that's where we

00:05:13.990 --> 00:05:15.750
get our dx, and of course we
sum them all up and that's

00:05:15.750 --> 00:05:18.510
where we get the integral.

00:05:18.510 --> 00:05:20.520
I'm going to do it as
two separate integrals.

00:05:20.520 --> 00:05:22.870
Some people will put them both
within the same integral, but I

00:05:22.870 --> 00:05:26.420
really want to hit the point
home that this is the volume of

00:05:26.420 --> 00:05:30.410
the outside surface, formed by
the outside surface or the cup,

00:05:30.410 --> 00:05:35.800
minus the volume formed by
the the inside function.

00:05:35.800 --> 00:05:42.840
It's going to be minus pi--
still going to be from 0 to 1--

00:05:42.840 --> 00:05:45.950
I drew fairly huge integral
signs, I don't know why-- and

00:05:45.950 --> 00:05:46.655
what's the inside function?

00:05:46.655 --> 00:05:49.740
It's x squared, and that's
going to be the radius of its

00:05:49.740 --> 00:05:54.030
own disks, if that's the
radius that's the disk, dx

00:05:54.030 --> 00:06:01.520
is the width, so it'll be x
squared, squared, times dx.

00:06:01.520 --> 00:06:02.800
So let's see if we
can figure that out.

00:06:02.800 --> 00:06:06.940
So, volume equals--
let's take the pis out.

00:06:06.940 --> 00:06:09.290
I think you'll see that that
pi is applying to everything,

00:06:09.290 --> 00:06:12.920
so we can take the pi out.

00:06:12.920 --> 00:06:19.310
And then, times the integral,
and now we can merge them

00:06:19.310 --> 00:06:24.170
back, because integrals
are additive like that.

00:06:24.170 --> 00:06:25.600
You'll see what I'm
talking about.

00:06:25.600 --> 00:06:26.720
This integral is
the same thing.

00:06:26.720 --> 00:06:28.090
So what's square
root of x squared?

00:06:28.090 --> 00:06:30.800
Well that's just x.

00:06:30.800 --> 00:06:33.390
And what's x squared, squared?

00:06:33.390 --> 00:06:34.780
That's x to the fourth, right?

00:06:34.780 --> 00:06:36.760
You multiply the exponents,
exponent rules.

00:06:36.760 --> 00:06:38.520
We have a minus sign here.

00:06:38.520 --> 00:06:45.070
Minus x to the fourth, all that
times dx, we've got that pi on

00:06:45.070 --> 00:06:49.430
the outside, that equals--
let's keep our pi

00:06:49.430 --> 00:06:51.040
on the outside.

00:06:51.040 --> 00:06:54.310
We're going to have to evaluate
the antiderivative at 1 and 0.

00:06:54.310 --> 00:06:57.360
So what's the
antiderivative of x?

00:06:57.360 --> 00:07:07.230
Well, that's x squared over
2, minus-- what's the

00:07:07.230 --> 00:07:08.730
antiderivative of
x to the fourth?

00:07:08.730 --> 00:07:11.140
Well, it's x to
the fifth over 5.

00:07:11.140 --> 00:07:14.010
That's hopefully
second nature to you.

00:07:14.010 --> 00:07:18.640
x to the fifth over 5, and
we're going to evaluate

00:07:18.640 --> 00:07:20.530
that at 1 and 0.

00:07:20.530 --> 00:07:21.122
1 and 0.

00:07:21.122 --> 00:07:23.100
We're going to subtract them.

00:07:23.100 --> 00:07:25.260
Fundamental theorem
of calculus.

00:07:25.260 --> 00:07:27.700
So that equals-- I'm going to
switch colors to avoid

00:07:27.700 --> 00:07:34.520
monotony-- that equals pi
times, let's evaluate it at 1,

00:07:34.520 --> 00:07:42.960
so it's 1/2 minus 1/5 and when
you evaluate it at 0, it's 0

00:07:42.960 --> 00:07:46.710
minus 0, so when you evaluate
0, you get nothing.

00:07:46.710 --> 00:07:48.940
And so what's 1/2 minus 1/5?

00:07:48.940 --> 00:07:57.250
That's pi times 2, get a common
denominator of 10, 5/2 is 1/2

00:07:57.250 --> 00:08:03.690
minus 2/10 is 1/5, so this
equals, this would be 3,

00:08:03.690 --> 00:08:07.190
so we get 3pi over 10.

00:08:07.190 --> 00:08:08.710
That's the volume formed.

00:08:08.710 --> 00:08:11.660
So it's almost easier to
figure out the volume of

00:08:11.660 --> 00:08:13.660
this figure than to draw it.

00:08:13.660 --> 00:08:16.670
Anyway, I think I'll leave you
there with this video, and in

00:08:16.670 --> 00:08:18.610
the next video, we're
going to start rotating

00:08:18.610 --> 00:08:19.510
around the y-axis.

00:08:19.510 --> 00:08:21.110
See you soon.


Solid of Revolution (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:01.150
Welcome back.

00:00:01.150 --> 00:00:03.010
I don't know what
I was thinking.

00:00:03.010 --> 00:00:04.420
Sometimes my brain
malfunctions.

00:00:04.420 --> 00:00:06.160
But just going back to that
problem we were doing, actually

00:00:06.160 --> 00:00:07.970
I think we should do it.

00:00:07.970 --> 00:00:09.960
I'm a little
schizophrenic today.

00:00:09.960 --> 00:00:12.210
So let's figure out
the equation for the

00:00:12.210 --> 00:00:14.220
volume of a sphere.

00:00:14.220 --> 00:00:15.630
So what's the equation?

00:00:15.630 --> 00:00:20.900
It's x squared plus y squared
is equal to r squared.

00:00:20.900 --> 00:00:23.630
And let's just write y as
a function of x, just so

00:00:23.630 --> 00:00:25.370
we can do it the way we
did that last problem.

00:00:25.370 --> 00:00:31.200
So you get y squared is equal
to r squared minus x squared.

00:00:31.200 --> 00:00:35.800
y is equal to the square root
of r squared minus x squared.

00:00:35.800 --> 00:00:37.410
And let's draw it.

00:00:37.410 --> 00:00:45.190
So if this is my y-axis, this
is my x-axis, and the

00:00:45.190 --> 00:00:52.420
equation-- draw it straight--
that's my x-axis, and then I

00:00:52.420 --> 00:00:55.190
actually have a circle tool,
let me see if I can use it

00:00:55.190 --> 00:01:10.760
effectively-- well,
close enough.

00:01:10.760 --> 00:01:11.190
There you go.

00:01:11.190 --> 00:01:12.390
I think you get the point.

00:01:12.390 --> 00:01:13.360
But anyway.

00:01:13.360 --> 00:01:16.570
This is just going to be the
upper half of the circle.

00:01:16.570 --> 00:01:18.330
Actually, I should probably
just undo that circle tool

00:01:18.330 --> 00:01:19.590
and try to draw it by hand.

00:01:19.590 --> 00:01:21.950
So y equals the square root of
r squared minus x squared.

00:01:21.950 --> 00:01:24.470
That's just going to be the
upper half of the circle.

00:01:24.470 --> 00:01:34.000
So it will be the positive x
quadrant-- and then actually,

00:01:34.000 --> 00:01:36.200
I should have drawn
the whole hemisphere.

00:01:36.200 --> 00:01:38.350
But anyway.

00:01:38.350 --> 00:01:39.970
Actually, let me do that,
because I think it'll

00:01:39.970 --> 00:01:46.140
make-- edit, undo, let me
clear all of this out.

00:01:46.140 --> 00:01:50.630
Sorry for wasting your time,
but I think it'll be effective.

00:01:50.630 --> 00:01:50.990
OK.

00:01:50.990 --> 00:01:51.775
So let me redraw.

00:01:54.460 --> 00:02:03.720
So this, that's the y-axis,
that's my x-axis, and then

00:02:03.720 --> 00:02:07.340
this-- the square root is,
since it's a function, it can

00:02:07.340 --> 00:02:10.020
only have one value, so we
assume it's defined as the

00:02:10.020 --> 00:02:11.500
positive square root.

00:02:11.500 --> 00:02:16.490
So if we were to graph that,
it would look like this.

00:02:16.490 --> 00:02:21.590
Something like that, where this
would be minus r and that's r.

00:02:21.590 --> 00:02:24.910
So if we want to find the
volume of a sphere with radius

00:02:24.910 --> 00:02:29.910
r, we just have to rotate this
function around the x-axis.

00:02:29.910 --> 00:02:33.740
This is the x-axis,
that's the y-axis.

00:02:33.740 --> 00:02:36.330
So let's see what we can do.

00:02:36.330 --> 00:02:38.690
Let's just visualize
the disks again.

00:02:38.690 --> 00:02:41.850
So let me make a disk.

00:02:41.850 --> 00:02:49.790
So let's say that that's the
side of one of the disks again,

00:02:49.790 --> 00:02:52.670
and as we know, the depth of
the disk is just

00:02:52.670 --> 00:02:54.590
going to be dx.

00:02:54.590 --> 00:02:58.000
That's how wide
that disk is, dx.

00:02:58.000 --> 00:03:01.210
And its radius at any point is
f of x, and in this case, it's

00:03:01.210 --> 00:03:04.180
y is equal to square root of
r squared minus x squared.

00:03:04.180 --> 00:03:07.740
So what's the surface
area of each disk?

00:03:07.740 --> 00:03:08.880
What's this?

00:03:08.880 --> 00:03:10.900
The surface area of
each of the disks.

00:03:10.900 --> 00:03:12.420
I hope you know
what I'm saying.

00:03:12.420 --> 00:03:16.850
So area is equal to pi r
squared, the radius at any

00:03:16.850 --> 00:03:20.490
point is equal to this, radius
is equal to y which is equal to

00:03:20.490 --> 00:03:23.770
square root-- and remember,
this is not this r.

00:03:23.770 --> 00:03:25.520
This is the radius
of this disk.

00:03:25.520 --> 00:03:26.500
I know it might be a
little confusing.

00:03:29.360 --> 00:03:34.940
y is equal to the square root
of r squared minus x squared.

00:03:34.940 --> 00:03:38.680
So the area is going to equal
pi times this squared.

00:03:38.680 --> 00:03:40.650
So if you square this quantity,
you just get rid of the

00:03:40.650 --> 00:03:42.560
square root sign, right?

00:03:42.560 --> 00:03:47.600
So pi r squared minus x
squared, and that's the

00:03:47.600 --> 00:03:50.020
area, and so what's the
volume of that disk?

00:03:50.020 --> 00:03:52.900
Well just like we've done in
every video up to this point,

00:03:52.900 --> 00:03:56.350
the volume of that disk is just
that, so the volume of that

00:03:56.350 --> 00:04:05.470
disk is just this pi r squared
minus x squared times dx.

00:04:05.470 --> 00:04:07.860
And so if we want to figure out
the volume of all these disks,

00:04:07.860 --> 00:04:10.255
I have a disk here, a disk
here, going around and around

00:04:10.255 --> 00:04:12.330
and around and around and
around and they get smaller and

00:04:12.330 --> 00:04:14.340
smaller until we have a sphere.

00:04:14.340 --> 00:04:19.020
We just take the integral, the
upper bound is positive r, the

00:04:19.020 --> 00:04:22.940
lower bound is minus r, and we
take the integral of

00:04:22.940 --> 00:04:24.980
this expression.

00:04:24.980 --> 00:04:27.240
pi-- let me distribute it,
because that's going to make it

00:04:27.240 --> 00:04:37.090
easier-- pi r squared, which is
just a constant term, minus pi

00:04:37.090 --> 00:04:41.560
x squared, all of that dx.

00:04:41.560 --> 00:04:43.870
So what's the antiderivative
of that expression?

00:04:43.870 --> 00:04:47.270
The antiderivative
within the parentheses.

00:04:47.270 --> 00:04:49.630
Well, this is just
a constant term.

00:04:49.630 --> 00:04:52.090
pi r squared, that's just a
number, because we're just

00:04:52.090 --> 00:04:57.550
taking the integral
with respect to x.

00:04:57.550 --> 00:05:04.660
So the antiderivative of pi r
squared is just pi r squared x,

00:05:04.660 --> 00:05:08.370
the derivative of pi r squared
x is just pi r squared, minus--

00:05:08.370 --> 00:05:11.160
and we did this in
the last video.

00:05:11.160 --> 00:05:13.670
Actually, well now, it's the
antiderivative x squared, which

00:05:13.670 --> 00:05:18.860
is x to the third over 3, and
the pi is just a constant, so

00:05:18.860 --> 00:05:25.670
pi x to the third over 3, and
we're going to evaluate

00:05:25.670 --> 00:05:29.440
that at r and minus r.

00:05:29.440 --> 00:05:31.940
Let me erase some stuff, looks
like I'm running out of space.

00:05:37.690 --> 00:05:40.840
Hopefully all of that
you know by now.

00:05:40.840 --> 00:05:45.620
OK, back to the pen tool.

00:05:45.620 --> 00:05:45.910
OK.

00:05:45.910 --> 00:05:49.080
So let's evaluate it at r.

00:05:49.080 --> 00:05:58.920
So this is pi r squared, and
then for x, we'll substitute

00:05:58.920 --> 00:06:07.440
the positive r times r minus pi
x cubed, but now we have this r

00:06:07.440 --> 00:06:21.870
here, so r cubed over 3 minus
pi r squared, and then we have

00:06:21.870 --> 00:06:25.290
a minus r here, because we're
evaluating the antiderivative

00:06:25.290 --> 00:06:36.120
at minus r, times minus r,
minus pi minus r cubed.

00:06:36.120 --> 00:06:37.240
So what's minus r cubed?

00:06:37.240 --> 00:06:40.370
It's r cubed, but we'll
keep the minus sign.

00:06:40.370 --> 00:06:42.540
r cubed, and at that minus
sign, let's just make

00:06:42.540 --> 00:06:46.650
that-- that'll turn that
into a plus-- over 3.

00:06:46.650 --> 00:06:49.070
Let's see if we can clean
this up a little bit.

00:06:49.070 --> 00:07:01.300
So that first term is pi r
cubed, r squared times r, minus

00:07:01.300 --> 00:07:08.780
essentially 1/3pi r cubed.

00:07:08.780 --> 00:07:10.310
And then, what is this?

00:07:10.310 --> 00:07:14.410
This is pi r cubed, but then
we have a minus sign up.

00:07:14.410 --> 00:07:17.090
This is minus pi r cubed, and
then we have a minus sign up

00:07:17.090 --> 00:07:28.310
here, so this becomes plus pi r
cubed, and then minus-- because

00:07:28.310 --> 00:07:30.830
we have a plus here and a minus
out here, so distribute it--

00:07:30.830 --> 00:07:37.610
so minus 1/3pi r cubed.

00:07:37.610 --> 00:07:39.420
And let's see, what do we have?

00:07:39.420 --> 00:07:41.680
We have essentially 1.

00:07:41.680 --> 00:07:48.780
If we just distribute out the
pi r cubes, we have pi r cubed

00:07:48.780 --> 00:07:57.200
times 1 minus 1/3
plus 1 minus 1/3.

00:07:57.200 --> 00:08:03.820
Well that's 2 minus 2/3, or
another way, let's see, is 2

00:08:03.820 --> 00:08:07.120
minus 2/3-- this is turning
into a fractions problem-- and

00:08:07.120 --> 00:08:11.970
what's-- well, that's 6/3
minus 2/3, it equals 4/3.

00:08:11.970 --> 00:08:14.280
So this is equal to 4/3.

00:08:14.280 --> 00:08:23.780
So the volume of the sphere is
equal to 4/3 pi r cubed, which

00:08:23.780 --> 00:08:28.240
is the equation for the
volume of a sphere.

00:08:28.240 --> 00:08:29.760
And actually, now that
I realize, it did take

00:08:29.760 --> 00:08:32.190
me eight and a half
minutes, so I am glad.

00:08:32.190 --> 00:08:35.310
My first intuition is always
correct, I am glad I did

00:08:35.310 --> 00:08:36.170
this in a separate video.

00:08:36.170 --> 00:08:37.650
But that should be pretty
interesting to you.

00:08:37.650 --> 00:08:40.840
And it makes a lot of sense.

00:08:40.840 --> 00:08:43.800
It's going to be a cube of
the radius, pi is involved.

00:08:43.800 --> 00:08:46.950
The 4/3 is interesting, just
in terms of how it relates

00:08:46.950 --> 00:08:49.400
to everything else.

00:08:49.400 --> 00:08:52.950
Area is pi r squared, and then
all of a sudden you get a 4/3

00:08:52.950 --> 00:08:54.360
here, so it is something
for you to think about.

00:08:54.360 --> 00:08:56.130
Anyway, hopefully
you found that fun.

00:08:56.130 --> 00:08:56.930
I'll see you in the next video.


Solid of Revolution  (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:01.420
Welcome back.

00:00:01.420 --> 00:00:04.610
On the last video we came to
the conclusion that we could

00:00:04.610 --> 00:00:08.020
figure out the volume when we
rotate a function about the

00:00:08.020 --> 00:00:12.400
x-axis, so let's apply that
to an actual exercise.

00:00:12.400 --> 00:00:14.500
I'm going to erase everything
because I don't want you to

00:00:14.500 --> 00:00:17.660
memorize this, because frankly
I haven't memorized this.

00:00:17.660 --> 00:00:21.170
And if you do, you'll forget
it one day and then you

00:00:21.170 --> 00:00:21.990
won't know how to do it.

00:00:21.990 --> 00:00:24.990
But if you understand why it
works then you'll never forget.

00:00:24.990 --> 00:00:26.550
As long as you remember
basic integration.

00:00:30.610 --> 00:00:33.490
Maybe you want to memorize it
if your teacher tends to give

00:00:33.490 --> 00:00:38.830
you a test that don't have
much extra time in them, just

00:00:38.830 --> 00:00:39.900
to speed up the process.

00:00:39.900 --> 00:00:41.720
But you should know
what's going on.

00:00:41.720 --> 00:00:44.090
So let me draw the axes again.

00:00:44.090 --> 00:00:46.340
That's my y-axis.

00:00:46.340 --> 00:00:47.420
That's my x-axis.

00:00:47.420 --> 00:00:49.708
And so since our first example
was y equals square root of

00:00:49.708 --> 00:00:51.235
x, let's stick with that.

00:00:51.235 --> 00:00:54.970
And for reasons that might
become apparent that tends to

00:00:54.970 --> 00:00:57.230
be one of the more typical
examples when you rotate

00:00:57.230 --> 00:00:58.980
things around axes.

00:00:58.980 --> 00:01:04.040
So let me see if I can draw it
as well as I drew it last time.

00:01:04.040 --> 00:01:05.230
Almost.

00:01:05.230 --> 00:01:07.920
OK, so that's y equals square
root of x, it's just f of x

00:01:07.920 --> 00:01:09.500
this time, I've defined it.

00:01:09.500 --> 00:01:11.110
This is the x-axis.

00:01:11.110 --> 00:01:12.980
That's the y-axis.

00:01:12.980 --> 00:01:16.000
And I'm going to rotate this
around the x-axis again.

00:01:16.000 --> 00:01:18.500
So I'm going to get a
sideways looking cup thing.

00:01:18.500 --> 00:01:20.780
And let's say I want to figure
out the volume of that cup

00:01:20.780 --> 00:01:24.680
between the points 0-- and to
make it simple, let's just

00:01:24.680 --> 00:01:30.370
say the points 0 and 1.

00:01:30.370 --> 00:01:32.670
So essentially we're just going
to get a cup, a sideways

00:01:32.670 --> 00:01:34.375
cup, it's going to look
something like this.

00:01:36.900 --> 00:01:38.280
It's going to look
something like this.

00:01:38.280 --> 00:01:42.590
That's going to be the--
that's a horrible-- the

00:01:42.590 --> 00:01:43.950
opening of the cup.

00:01:43.950 --> 00:01:46.490
Actually why don't I
use the circle tool.

00:01:46.490 --> 00:01:48.650
It just dawned on me that
I had a circle tool.

00:01:48.650 --> 00:01:53.255
So the opening of the cup
will look like that.

00:01:55.975 --> 00:01:58.210
Actually I could
draw it right here.

00:01:58.210 --> 00:02:01.020
This would be the
opening of the cup.

00:02:01.020 --> 00:02:05.820
Well-- you can see sometimes
that my videos are

00:02:05.820 --> 00:02:07.980
a little unplanned.

00:02:07.980 --> 00:02:09.650
There you go.

00:02:09.650 --> 00:02:12.230
So that would be the
opening of the cup.

00:02:12.230 --> 00:02:14.320
This is excellent.

00:02:14.320 --> 00:02:18.550
This tool is very well suited
for what I'm doing here.

00:02:18.550 --> 00:02:20.390
We're rotating around that way.

00:02:20.390 --> 00:02:21.280
We're turning that function.

00:02:21.280 --> 00:02:22.980
So the cup's going to look like
that, so the bottom part of the

00:02:22.980 --> 00:02:25.390
cup's going to look like this.

00:02:25.390 --> 00:02:28.310
And it's solid, so we want the
volume of the whole thing.

00:02:28.310 --> 00:02:30.310
In future videos I'm going to
show you actually how to figure

00:02:30.310 --> 00:02:32.770
out the surface area of the
cup, which I find in some

00:02:32.770 --> 00:02:33.950
ways more interesting.

00:02:33.950 --> 00:02:35.250
So how do we think
about that again?

00:02:35.250 --> 00:02:37.220
Let's just rederive it,
but this time we'll use

00:02:37.220 --> 00:02:38.610
a specific equation.

00:02:38.610 --> 00:02:41.710
So we just have to figure out
what is the volume of one disk

00:02:41.710 --> 00:02:43.420
and then sum up all the disks.

00:02:43.420 --> 00:02:47.970
So let's say this disk right
here-- actually let's just take

00:02:47.970 --> 00:02:50.090
this disk at the end point
right here that I've already

00:02:50.090 --> 00:02:52.850
drawn something for.

00:02:52.850 --> 00:02:56.500
So what's the radius
of this disk?

00:02:56.500 --> 00:02:59.330
The radius of that disk
is f of x at that point.

00:02:59.330 --> 00:03:03.830
Well f of x at that point
is just square root of x.

00:03:03.830 --> 00:03:07.740
Radius is equal to
square root of x.

00:03:07.740 --> 00:03:17.870
And so the area of that disk is
going to equal pi r squared.

00:03:17.870 --> 00:03:20.830
Well the radius is square root
of x, so it equals pi times

00:03:20.830 --> 00:03:22.540
square root of x squared.

00:03:22.540 --> 00:03:26.440
So it equals pi times x.

00:03:26.440 --> 00:03:28.830
That's the area of each disk.

00:03:28.830 --> 00:03:32.860
And then if we want the volume,
you just have to multiply the

00:03:32.860 --> 00:03:36.550
area of that surface times
the depth of the disk.

00:03:36.550 --> 00:03:37.870
I'm just trying to show.

00:03:37.870 --> 00:03:39.530
You can imagine that this is
kind of like a quarter and this

00:03:39.530 --> 00:03:40.600
is the side of the quarter.

00:03:43.590 --> 00:03:46.730
We saw in the last video that
depth, that's just a very

00:03:46.730 --> 00:03:50.710
small change in x, because
we want each disk to be

00:03:50.710 --> 00:03:53.540
infinitesimally thin.

00:03:53.540 --> 00:03:57.720
So the width is just
dx at any point.

00:03:57.720 --> 00:04:03.550
So the volume of each disk is
equal to the area, which we

00:04:03.550 --> 00:04:10.305
just figured out was pi x
times the depth, times dx.

00:04:10.305 --> 00:04:12.190
That's the volume of each disk.

00:04:12.190 --> 00:04:17.620
So the total volume is
going to be equal to the

00:04:17.620 --> 00:04:18.840
sum of all of these.

00:04:18.840 --> 00:04:20.470
That was one disk I drew, then
you're going to have another

00:04:20.470 --> 00:04:22.980
one here, you're going to have
another one here,

00:04:22.980 --> 00:04:23.610
another one here.

00:04:23.610 --> 00:04:25.100
You're going to have infinitely
many, and you want them to be

00:04:25.100 --> 00:04:27.250
super, super, super thin so
that you get an accurate

00:04:27.250 --> 00:04:30.120
measure of the exact
volume of this curve.

00:04:30.120 --> 00:04:31.890
Otherwise it would just be an
approximation, and that's

00:04:31.890 --> 00:04:33.330
where we use the integral.

00:04:33.330 --> 00:04:34.900
So it will be the
integral from.

00:04:34.900 --> 00:04:41.170
And my original
boundaries were 0 to 1.

00:04:41.170 --> 00:04:43.600
The disk we used as an example,
this is probably you know the

00:04:43.600 --> 00:04:49.290
last disk, so this one will
actually have a radius of the

00:04:49.290 --> 00:04:52.250
square root of 1, which is 1.

00:04:52.250 --> 00:04:54.840
Not that you have to know that,
I'm just trying to keep

00:04:54.840 --> 00:04:57.730
emphasizing the visualization.

00:04:57.730 --> 00:04:59.240
So what will be the integral?

00:04:59.240 --> 00:05:03.500
Well we're going to go from 0
to 1, and we're going to sum up

00:05:03.500 --> 00:05:05.820
a bunch of these disks, which
we've already defined,

00:05:05.820 --> 00:05:09.530
so it's pi x dx.

00:05:09.530 --> 00:05:13.010
This is looking to be a fairly
straightforward integral.

00:05:13.010 --> 00:05:14.300
So what's the integral of that?

00:05:20.400 --> 00:05:25.670
Pi is just a constant and the
antiderivative of x is x to

00:05:25.670 --> 00:05:27.525
the 1/2 over-- I'm sorry.

00:05:30.150 --> 00:05:31.795
It's x squared over 2.

00:05:34.980 --> 00:05:36.840
I've been a little rusty
since I last did some

00:05:36.840 --> 00:05:38.410
antiderivatives.

00:05:38.410 --> 00:05:45.330
So we get x squared, we get
pi times x squared over 2.

00:05:48.420 --> 00:05:50.460
That's the
antiderivative of that.

00:05:50.460 --> 00:05:53.536
And then we have to evaluate
it at 1 and then subtract

00:05:53.536 --> 00:05:56.010
it and evaluate it at 0.

00:05:56.010 --> 00:05:58.860
And so what do we have.

00:05:58.860 --> 00:06:07.960
We get 1/2 pi, so we get pi
over 2 minus 0 pi, minus 0.

00:06:07.960 --> 00:06:11.950
So it equals pi over 2.

00:06:11.950 --> 00:06:12.510
There we go.

00:06:12.510 --> 00:06:17.610
We just figured out the volume
of this cup from 0 to 1.

00:06:17.610 --> 00:06:19.730
Reasonably interesting.

00:06:19.730 --> 00:06:23.210
Let's see if we can do that
again to figure out the-- just

00:06:23.210 --> 00:06:28.880
to give you another example,
just hit the point home-- to

00:06:28.880 --> 00:06:30.460
see if we can figure out the
volume of a sphere,

00:06:30.460 --> 00:06:33.340
the equation for the
volume of a sphere.

00:06:33.340 --> 00:06:36.480
So what's the equation
for a circle?

00:06:36.480 --> 00:06:43.770
It's x squared plus y squared
is equal to r squared.

00:06:43.770 --> 00:06:46.540
And let's write that in terms
of y is a function of x, just

00:06:46.540 --> 00:06:48.530
so we have something that
we can work with the

00:06:48.530 --> 00:06:49.260
way we learned it.

00:06:49.260 --> 00:06:59.300
So we get y squared is equal to
r squared minus x squared, and

00:06:59.300 --> 00:07:09.600
then we get y is equal to the
square root of r squared

00:07:09.600 --> 00:07:12.380
minus x squared.

00:07:12.380 --> 00:07:15.920
Actually now that I realize it,
I'm going to not do this,

00:07:15.920 --> 00:07:19.910
because I think I'm going into
too complicated a problem.

00:07:19.910 --> 00:07:21.910
I did that on a fly.

00:07:21.910 --> 00:07:25.040
But in the next video I will
do slightly more complicated

00:07:25.040 --> 00:07:26.640
without going to this one,
because I probably don't have

00:07:26.640 --> 00:07:28.100
time for it I just realized.

00:07:28.100 --> 00:07:29.350
Anyway I'll see you
in the next video.


Solid of Revolution (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.220
Well we already know that
definite integrals can help us

00:00:03.220 --> 00:00:07.240
figure out areas underneath
curves or between curves.

00:00:07.240 --> 00:00:09.930
What we'll show in this video
is that you can actually use

00:00:09.930 --> 00:00:12.950
pretty much the exact same
principles to figure out the

00:00:12.950 --> 00:00:15.580
volumes of rotational solids.

00:00:15.580 --> 00:00:16.490
So what do I mean?

00:00:16.490 --> 00:00:19.060
So let me just draw a
couple of examples.

00:00:19.060 --> 00:00:26.220
So let me start with a fairly
straightforward function.

00:00:26.220 --> 00:00:28.980
That's my y-axis.

00:00:28.980 --> 00:00:31.930
This is my x-axis.

00:00:31.930 --> 00:00:32.885
Let me draw my function.

00:00:35.840 --> 00:00:38.500
I'm going to draw y equals the
square root of x, but I'll keep

00:00:38.500 --> 00:00:41.090
it general right now, just so
this applies generally

00:00:41.090 --> 00:00:42.210
to a lot of things.

00:00:42.210 --> 00:00:46.560
So y equals the square root of
x looks something like that.

00:00:46.560 --> 00:00:47.590
It keeps going.

00:00:47.590 --> 00:00:49.685
Actually, let me redraw that,
because I didn't want it

00:00:49.685 --> 00:00:51.990
to curve down like
that at the end.

00:00:51.990 --> 00:00:54.410
It goes up like that,
and then it just keeps

00:00:54.410 --> 00:00:55.260
going up like that.

00:00:55.260 --> 00:00:56.210
That's better.

00:00:56.210 --> 00:00:56.980
OK.

00:00:56.980 --> 00:01:00.910
So we'll call that f of x.

00:01:00.910 --> 00:01:02.900
This is our x-axis.

00:01:02.900 --> 00:01:05.220
This is our y-axis.

00:01:05.220 --> 00:01:07.390
And we already know that if
we wanted to figure out

00:01:07.390 --> 00:01:10.980
the area under this curve
between two points.

00:01:10.980 --> 00:01:15.445
Let's say between the point
a-- well, we can do it

00:01:15.445 --> 00:01:16.060
between any two points.

00:01:16.060 --> 00:01:20.950
Let's say this point a and this
point b, and we wanted to find

00:01:20.950 --> 00:01:22.390
this area between
the two curves.

00:01:24.925 --> 00:01:26.700
I'm drawing everything
crooked today.

00:01:26.700 --> 00:01:27.980
Let me do a different color.

00:01:33.730 --> 00:01:37.990
If we wanted this area right
here, we would essentially just

00:01:37.990 --> 00:01:40.690
be-- just as a review-- be
summing up a bunch of small

00:01:40.690 --> 00:01:45.060
squares, where each square has
a bunch of rectangles, has a

00:01:45.060 --> 00:01:50.240
width dx, and its height at
that point would be whatever x

00:01:50.240 --> 00:01:52.530
value here is-- it
would be f of x.

00:01:52.530 --> 00:01:56.100
And if we take the sum of all
of these areas, of all of these

00:01:56.100 --> 00:01:57.990
rectangles, we would get
the area to this curve.

00:01:57.990 --> 00:02:01.430
And we learned in the definite
integral video that that's just

00:02:01.430 --> 00:02:05.930
equal to the definite integral
from a-- that's a lower bound--

00:02:05.930 --> 00:02:13.440
from a to b of f of
x times d of x.

00:02:13.440 --> 00:02:14.100
Where each [? rafter ?]

00:02:14.100 --> 00:02:15.810
angle is f of x times d of x.

00:02:15.810 --> 00:02:18.030
And hopefully this makes a
little bit of intuitive

00:02:18.030 --> 00:02:18.480
sense to you.

00:02:18.480 --> 00:02:21.370
I think a lot of people go
through calculus just learning

00:02:21.370 --> 00:02:25.160
how to do it mechanistically,
just learning you know how to

00:02:25.160 --> 00:02:27.040
do it like a robot without
really understanding

00:02:27.040 --> 00:02:28.320
necessarily what's going on.

00:02:28.320 --> 00:02:30.180
And if you understand what's
going on, you'll never be

00:02:30.180 --> 00:02:34.210
really lost when you see
something a little bit

00:02:34.210 --> 00:02:36.170
different than what you
might have practiced.

00:02:36.170 --> 00:02:40.340
So with that out of the way,
let's think about something.

00:02:40.340 --> 00:02:43.690
What if we took this
function and we rotated

00:02:43.690 --> 00:02:45.270
it about the x-axis.

00:02:45.270 --> 00:02:47.440
So this might take a little bit
of visualization, but what you

00:02:47.440 --> 00:02:49.650
imagine is-- let me
see if I can draw.

00:02:49.650 --> 00:02:56.370
So I take this curve and if I
were to rotate it about the

00:02:56.370 --> 00:02:58.370
x-axis, it would look
something like this.

00:03:02.800 --> 00:03:06.650
It would look like a
sideways copper vase.

00:03:09.152 --> 00:03:12.730
It would look like that, where
that would be the opening.

00:03:12.730 --> 00:03:15.080
That would be the
opening on the inside.

00:03:15.080 --> 00:03:16.570
I can even shade it.

00:03:16.570 --> 00:03:18.610
Show you my drawing skills.

00:03:18.610 --> 00:03:19.220
OK.

00:03:19.220 --> 00:03:20.890
Hopefully that make sense.

00:03:20.890 --> 00:03:25.580
You know, that would be the
y-axis there, and the x-axis

00:03:25.580 --> 00:03:28.050
would pop out the middle.

00:03:28.050 --> 00:03:31.000
That's if you took this and
you rotated it around.

00:03:31.000 --> 00:03:34.210
So let me draw an arrow to show
that we're rotating it around.

00:03:38.360 --> 00:03:41.980
And if we did that, what would
be the volume created by--

00:03:41.980 --> 00:03:44.830
well let's use the same
boundaries, between a and b.

00:03:44.830 --> 00:03:48.240
So if we took this piece and we
rotated it around, what would

00:03:48.240 --> 00:03:50.110
it look like between a and b.

00:03:50.110 --> 00:03:51.325
It would look
something like this.

00:03:54.410 --> 00:03:56.360
Let me see if I can draw it.

00:03:56.360 --> 00:04:04.910
So you would have a-- whoops,
I'm not trying that well.

00:04:04.910 --> 00:04:07.370
This is really testing the
limits of my ability to use

00:04:07.370 --> 00:04:08.750
this computer to draw things.

00:04:12.560 --> 00:04:17.700
It'd be kind of a circle on one
end, and then it would curve

00:04:17.700 --> 00:04:25.030
down a little bit and it would
be another circle

00:04:25.030 --> 00:04:26.180
on the other end.

00:04:26.180 --> 00:04:28.480
And if I were to draw the
x-axis, the x-axis would

00:04:28.480 --> 00:04:34.370
kind of pop out of the
middle right there.

00:04:34.370 --> 00:04:39.810
That right there would be
the point b, x equals b.

00:04:39.810 --> 00:04:44.340
If we were to kind of go behind
or look into the object we

00:04:44.340 --> 00:04:50.950
would see the other surface
of this rotational solid.

00:04:50.950 --> 00:04:52.925
And this point right
here, that would be a.

00:04:56.460 --> 00:04:57.520
That would be a.

00:04:57.520 --> 00:05:00.500
And then of course the x-axis
would keep going, and then

00:05:00.500 --> 00:05:01.570
that would be the y-axis.

00:05:04.150 --> 00:05:07.770
The visualization really
is the hardest part

00:05:07.770 --> 00:05:09.280
about these problems.

00:05:09.280 --> 00:05:14.620
So first just actually
imagine what you're doing.

00:05:14.620 --> 00:05:18.960
So I just did this section, if
I rotated it about the x-axis.

00:05:18.960 --> 00:05:21.680
But if I were to draw the whole
curve, the whole curve would

00:05:21.680 --> 00:05:24.920
look something like this.

00:05:24.920 --> 00:05:27.330
It would look something
like that, and we're

00:05:27.330 --> 00:05:28.410
just rotating it around.

00:05:28.410 --> 00:05:29.215
Hopefully that makes sense.

00:05:29.215 --> 00:05:31.810
We're rotating it
around that way.

00:05:31.810 --> 00:05:33.640
So how do we do that?

00:05:33.640 --> 00:05:35.540
Well we use the exact
same principle.

00:05:35.540 --> 00:05:38.870
When we figured out the area,
we would figure out the area of

00:05:38.870 --> 00:05:42.920
each of these small squares,
and then we would take the sum

00:05:42.920 --> 00:05:45.780
of an infinite number of
infinitely small squares,

00:05:45.780 --> 00:05:48.000
and we got this.

00:05:48.000 --> 00:05:51.220
So to do the volume, what we
do is instead of having each

00:05:51.220 --> 00:05:56.170
rectangle, we kind of rotate
each of these rectangles

00:05:56.170 --> 00:05:57.650
around the x-axis.

00:06:01.660 --> 00:06:10.920
If that's the rectangle,
it has width dx, and

00:06:10.920 --> 00:06:12.070
it has height f of x.

00:06:16.870 --> 00:06:22.240
So this height right here,
that's f of x at this point.

00:06:22.240 --> 00:06:27.930
If I were to rotate this
rectangle around the x-axis,

00:06:27.930 --> 00:06:29.860
what do I end up with?

00:06:29.860 --> 00:06:32.040
Well I'll end up with a disk.

00:06:32.040 --> 00:06:36.460
Let me see if I can draw
that reasonably well.

00:06:36.460 --> 00:06:40.430
I'm trying to show you some
perspective when I draw.

00:06:43.050 --> 00:06:46.670
So that would be the top
surface of the disk.

00:06:46.670 --> 00:06:49.030
And this would be the
side of the disk.

00:06:52.430 --> 00:06:56.010
And so this is the top
surface at the disk.

00:06:56.010 --> 00:06:58.360
And what would be the radius
of this disk, what would

00:06:58.360 --> 00:07:00.610
be this height right here?

00:07:00.610 --> 00:07:03.260
Well that radius, that's
going to be f of x.

00:07:03.260 --> 00:07:04.630
That's this height.

00:07:04.630 --> 00:07:08.310
Imagine if you took this and
rotated it around, that's the

00:07:08.310 --> 00:07:11.050
same thing as this height
right here, right?

00:07:11.050 --> 00:07:15.050
So that height or the radius
of the disk is f of x.

00:07:15.050 --> 00:07:18.230
And then what's the
width of the disk?

00:07:18.230 --> 00:07:20.350
Well that's just d of x.

00:07:20.350 --> 00:07:21.430
That's the same thing as this.

00:07:21.430 --> 00:07:23.080
We just rotated it around.

00:07:23.080 --> 00:07:25.390
So what would be the
volume of this disk?

00:07:25.390 --> 00:07:29.520
It would be the
area of this side.

00:07:29.520 --> 00:07:35.780
It'll be this area right
here times this height.

00:07:35.780 --> 00:07:36.930
Well what's the area?

00:07:36.930 --> 00:07:38.310
Well we know the radius, right?

00:07:38.310 --> 00:07:41.805
Area is equal to pi r squared.

00:07:44.920 --> 00:07:46.550
What's this radius?

00:07:46.550 --> 00:07:48.060
My radius is f of x, right?

00:07:48.060 --> 00:07:55.620
So the area of this disk is
equal to pi times the radius

00:07:55.620 --> 00:08:03.100
squared, it so it's pi times f
of x, the whole thing squared.

00:08:03.100 --> 00:08:06.010
So what would be the volume
of this entire disk?

00:08:06.010 --> 00:08:10.620
So it'll just be
this area times dx.

00:08:10.620 --> 00:08:12.890
I'm running out of
space and colors.

00:08:12.890 --> 00:08:19.700
So the volume of that disk is
going to be equal to area of

00:08:19.700 --> 00:08:25.010
that disk, pi f of x squared.

00:08:25.010 --> 00:08:27.500
The whole function, whatever
length this is at any point

00:08:27.500 --> 00:08:30.390
squared, that gives us the
area, times the depth you

00:08:30.390 --> 00:08:32.980
can say, so that's d of x.

00:08:32.980 --> 00:08:36.340
Now that gives us just the
volume of this one disk

00:08:36.340 --> 00:08:38.880
when it's rotated around.

00:08:38.880 --> 00:08:42.080
So if we wanted the volume of
this entire object that I drew

00:08:42.080 --> 00:08:45.900
here, we would just sum up
a bunch of these disks.

00:08:45.900 --> 00:08:49.910
We would take each of these
rectangles, rotate them around,

00:08:49.910 --> 00:08:52.370
figure out the volume of that
disk it creates, and

00:08:52.370 --> 00:08:53.780
then sum them up.

00:08:53.780 --> 00:08:57.980
And so essentially we're going
to take an infinite sum of a

00:08:57.980 --> 00:09:00.250
bunch of these small
little disks so we can

00:09:00.250 --> 00:09:01.380
take the integral.

00:09:01.380 --> 00:09:04.160
So this is the volume
of each disk.

00:09:04.160 --> 00:09:06.450
We could call that a
volume of a disk.

00:09:06.450 --> 00:09:07.720
So what's the volume
of the whole thing?

00:09:07.720 --> 00:09:11.700
Well we just take a sum,
an integral sum of

00:09:11.700 --> 00:09:13.110
each of these disks.

00:09:13.110 --> 00:09:19.040
So the volume when you rotate
it is going to be equal to the

00:09:19.040 --> 00:09:22.370
definite integral between-- and
remember, our boundaries were a

00:09:22.370 --> 00:09:29.920
and b-- between a and b of this
quantity right here--

00:09:29.920 --> 00:09:39.950
pi f of x squared dx.

00:09:39.950 --> 00:09:41.820
So hopefully that
makes sense to you.

00:09:41.820 --> 00:09:45.340
Just remember, this is
the width of each disk.

00:09:45.340 --> 00:09:48.900
This is the radius of the disk,
or the radius of the surface,

00:09:48.900 --> 00:09:50.555
so it would be squared, and
that makes sense, that's

00:09:50.555 --> 00:09:51.390
the height, f of x.

00:09:51.390 --> 00:09:54.520
And we have pi r squared, so
that's where the pi comes from.

00:09:54.520 --> 00:09:55.780
Some people just memorize that.

00:09:55.780 --> 00:09:58.140
I don't recommend you do that,
and we'll see that later.

00:09:58.140 --> 00:09:59.290
But I'm out of time.

00:09:59.290 --> 00:10:01.450
In the next video I'll
actually apply this

00:10:01.450 --> 00:10:02.685
to an actual problem.

00:10:02.685 --> 00:10:02.940
See you soon.


Thermodynamics part 5: Molar ideal gas law problem

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:03.810
I told you that the two most
important things you should

00:00:03.810 --> 00:00:07.860
know in thermodynamics that will
get you most of your way

00:00:07.860 --> 00:00:10.890
through most exams is that the
pressure times the volume is

00:00:10.890 --> 00:00:14.440
equal to a constant, and that
the pressure times the volume

00:00:14.440 --> 00:00:17.310
divided by the temperatures
is equal to a constant.

00:00:17.310 --> 00:00:22.200
They all change such that the
initial pressure times the

00:00:22.200 --> 00:00:25.490
volume divided by the initial
temperature is equal to the

00:00:25.490 --> 00:00:27.520
final pressure times volume
divided by the final

00:00:27.520 --> 00:00:27.830
temperature.

00:00:27.830 --> 00:00:29.380
Assuming that you're not
changing the energy of the

00:00:29.380 --> 00:00:31.060
system, and we'll do
more on that later.

00:00:31.060 --> 00:00:33.240
The other thing you should
remember is that pressure

00:00:33.240 --> 00:00:36.520
times volume is equal
to n, where n is

00:00:36.520 --> 00:00:37.970
the number of moles.

00:00:37.970 --> 00:00:40.930
Mole is a number like dozen, but
mole is a huge number-- 6

00:00:40.930 --> 00:00:44.960
times 10 to the 23 times R.

00:00:44.960 --> 00:01:02.440
R was the universal gas
constant-- that's 8.31 joules

00:01:02.440 --> 00:01:07.780
per mole Kelvin times
the temperature.

00:01:07.780 --> 00:01:11.610
Remember, just to be safe,
always convert to Kelvin

00:01:11.610 --> 00:01:12.440
first.

00:01:12.440 --> 00:01:15.430
Let's see if we can do a problem
that I can make up on

00:01:15.430 --> 00:01:17.290
the fly of this situation.

00:01:17.290 --> 00:01:24.170
Let's say I have a balloon, and
the volume of the balloon

00:01:24.170 --> 00:01:32.650
is 1 meter cubed, so this
is a big balloon.

00:01:32.650 --> 00:01:35.870
That's fairly large, if you
imagine a cubic meter.

00:01:35.870 --> 00:01:39.420
The volume is a cubic meter, and
let's say the pressure is

00:01:39.420 --> 00:01:50.240
equal to 5 pascals, and that's
newtons per meter squared.

00:01:53.105 --> 00:01:57.480
And let's say we're at a
reasonably warm temperature,

00:01:57.480 --> 00:02:04.780
so temperature is equal to 20
degrees Celsius, and let's say

00:02:04.780 --> 00:02:08.750
that balloon is filled helium.

00:02:08.750 --> 00:02:13.170
My question to you is, how many
molecules of helium do I

00:02:13.170 --> 00:02:14.490
have in the balloon?

00:02:14.490 --> 00:02:16.180
Let's just substitute
into the equation.

00:02:16.180 --> 00:02:20.150
We have pressure, which is 5--
and I'll actually write the

00:02:20.150 --> 00:02:21.670
units, I never do it, but you
should, and you should always

00:02:21.670 --> 00:02:25.460
do it on an exam-- 5
newtons per meter

00:02:25.460 --> 00:02:28.550
squared times the volume.

00:02:28.550 --> 00:02:34.330
1 meter cubed is equal to my
number of moles, n, times the

00:02:34.330 --> 00:02:44.590
universal gas constant, 8.31
joules per mole Kelvin, times

00:02:44.590 --> 00:02:45.610
temperature.

00:02:45.610 --> 00:02:49.210
Remember, and I can't repeat
this enough, always convert

00:02:49.210 --> 00:02:52.360
the temperature to Kelvin--
so whatever our Celsius

00:02:52.360 --> 00:02:55.000
temperature is, add 273.

00:02:55.000 --> 00:02:59.540
Add 273 to that, and
you get 293 Kelvin.

00:02:59.540 --> 00:03:04.720
So I get 5 times 1, and meters
square, meters cubed cancels

00:03:04.720 --> 00:03:05.760
out and just becomes a meter.

00:03:05.760 --> 00:03:07.290
Newton meter is joules.

00:03:07.290 --> 00:03:11.860
5 joules is equal to
n moles times 8.31

00:03:11.860 --> 00:03:14.490
joules per mole Kelvin.

00:03:14.490 --> 00:03:29.580
This Kelvin and this Kelvin
cancel out, so 8.31 times 293

00:03:29.580 --> 00:03:45.235
is equal to 2,434.83
joules per mole.

00:03:48.660 --> 00:03:51.380
To get to the number of moles,
we just divide both sides of

00:03:51.380 --> 00:03:52.870
this equation by that.

00:03:52.870 --> 00:03:56.570
And the units should work
out, so you get 5.

00:03:56.570 --> 00:04:10.950
So n is equal to 5 joules
times 1 over 2,434.83.

00:04:10.950 --> 00:04:15.740
Since we're dividing by this,
this flips; moles per joule.

00:04:15.740 --> 00:04:17.740
This joule cancels out with this
joule, so we just have to

00:04:17.740 --> 00:04:21.970
divide 5 by this, and we'll
get the number of moles.

00:04:21.970 --> 00:04:27.680
Let's take the inverse of what
I had there times 5, and so I

00:04:27.680 --> 00:04:32.552
get 0.002 moles.

00:04:32.552 --> 00:04:41.400
So this equals 0.0021 moles.

00:04:41.400 --> 00:04:44.100
That might seem like a small
number to you, but let's

00:04:44.100 --> 00:04:46.370
figure out how many
molecules that is.

00:04:49.750 --> 00:04:54.150
Let me make some space free,
so I can write Avogadro's

00:04:54.150 --> 00:04:55.400
number down.

00:05:06.730 --> 00:05:08.720
Did I even say what Avogadro's
number is?

00:05:08.720 --> 00:05:11.090
Avogadro's number is the number
of molecules per mole--

00:05:11.090 --> 00:05:12.010
it's that number.

00:05:12.010 --> 00:05:19.690
So, number Avogadro is equal
to 6.022 times 10 to the 23

00:05:19.690 --> 00:05:24.040
molecules per mole.

00:05:24.040 --> 00:05:25.870
The top is molecules, and the
bottom is moles-- I know you

00:05:25.870 --> 00:05:27.390
can't read that.

00:05:27.390 --> 00:05:31.970
I have 0.0021 moles, so how
many molecules do I have?

00:05:31.970 --> 00:05:38.750
I just multiply that 0.0021
times how many moles per

00:05:38.750 --> 00:05:45.580
molecule-- because this is
moles-- times Avogadro's

00:05:45.580 --> 00:05:52.430
number, which is molecules
per mole.

00:05:52.430 --> 00:05:54.910
That's molecules, this is
moles-- maybe I should write

00:05:54.910 --> 00:05:59.530
the whole thing-- so then the
moles cancel out, and

00:05:59.530 --> 00:06:08.070
Avogadro's number is 6.022 times
10 to the 23-- let's

00:06:08.070 --> 00:06:09.900
just remember that--
and let's just

00:06:09.900 --> 00:06:16.650
multiply that times 0.0021.

00:06:16.650 --> 00:06:32.880
It equals 0.0126 times 10
to the 23 molecules.

00:06:32.880 --> 00:06:36.240
This is 0.0126.

00:06:36.240 --> 00:06:48.260
That's the same thing as 1.26
times 0.01, and then of course

00:06:48.260 --> 00:06:50.740
times 10 to the 23.

00:06:50.740 --> 00:06:51.710
What's 0.01?

00:06:51.710 --> 00:06:56.320
That's 10 to the negative 2--
10 to the negative 1 is 0.1,

00:06:56.320 --> 00:06:59.110
so this 10 to the negative 2.

00:06:59.110 --> 00:07:04.410
Then we get 1.26-- 10 to the
negative 2, times 10 to the

00:07:04.410 --> 00:07:09.760
negative 3, and we add the
exponents times 10 to the 21.

00:07:09.760 --> 00:07:14.790
It's roughly 1.26, and then
another 19 zeroes-- or roughly

00:07:14.790 --> 00:07:19.360
1 followed by 21 zeroes-- is how
many molecules of, in this

00:07:19.360 --> 00:07:21.260
case, helium we had
in the balloon.

00:07:21.260 --> 00:07:22.230
It's not too difficult.

00:07:22.230 --> 00:07:24.950
The hard part is really just
remembering Avogadro's number,

00:07:24.950 --> 00:07:29.330
remembering the universal gas
constant is 8.31 joules per

00:07:29.330 --> 00:07:32.920
mole Kelvin, remembering
to always convert your

00:07:32.920 --> 00:07:35.300
temperature to Kelvin, and then
just making sure all your

00:07:35.300 --> 00:07:36.390
units match up.

00:07:36.390 --> 00:07:38.660
Sometimes that might be tricky:
they might give volume

00:07:38.660 --> 00:07:41.660
in liters, and you have to--
especially in this case--

00:07:41.660 --> 00:07:44.640
convert it to meters cubed
before you do it.

00:07:44.640 --> 00:07:47.830
They might give pressure,
atmospheres, or bars, and you

00:07:47.830 --> 00:07:50.980
should know the conversion, and
convert it to pascals or

00:07:50.980 --> 00:07:52.040
newtons per meter squared.

00:07:52.040 --> 00:07:54.600
Other than that, it's just
substituting and just doing

00:07:54.600 --> 00:07:57.210
the hairy math and the
scientific notation.

00:07:57.210 --> 00:07:59.600
Hopefully, that was vaguely
clarifying.

00:07:59.600 --> 00:08:01.400
See you in the next video.


Thermodynamics part 4: Moles and the ideal gas law

WEBVTT
Kind: captions
Language: en

00:00:01.050 --> 00:00:04.280
Before I continue, I want to
introduce you to what might be

00:00:04.280 --> 00:00:06.050
an unfamiliar concept, although
if you've taken

00:00:06.050 --> 00:00:08.710
chemistry, you might know a
little bit about it-- it's

00:00:08.710 --> 00:00:10.460
called a mole.

00:00:10.460 --> 00:00:13.030
This isn't the thing that grows
on your face with the

00:00:13.030 --> 00:00:16.950
hair in it, or the animal that
digs in your backyard,

00:00:16.950 --> 00:00:19.990
although those are also
called moles.

00:00:19.990 --> 00:00:22.540
We're talking about the
SI unit called a mole.

00:00:22.540 --> 00:00:24.390
A mole is just a number.

00:00:24.390 --> 00:00:28.010
It's like saying a mole of
something means a certain

00:00:28.010 --> 00:00:34.700
number of something, just like
a dozen-- it's like saying

00:00:34.700 --> 00:00:41.120
that a dozen eggs is 12 eggs.

00:00:41.120 --> 00:00:47.970
Just like that, a mole would
be 6-- I always forget the

00:00:47.970 --> 00:00:53.550
exact number, but it's 6.023,
or something of that nature.

00:00:53.550 --> 00:00:56.700
You could look it up--
I think it's 6.023

00:00:56.700 --> 00:00:57.920
times 10 to the twentieth.

00:00:57.920 --> 00:01:01.320
Let me look up the exact number,
just because I think

00:01:01.320 --> 00:01:06.840
that 23 I'm misremembering.

00:01:06.840 --> 00:01:24.730
The mole is 6.022 times 10 to
the 23 of something, so it's a

00:01:24.730 --> 00:01:27.870
very number of something.

00:01:27.870 --> 00:01:31.110
Normally, we don't deal in moles
of eggs-- I don't think

00:01:31.110 --> 00:01:34.870
there has been a mole of eggs
ever produced in the history

00:01:34.870 --> 00:01:35.390
of the universe.

00:01:35.390 --> 00:01:40.220
10 to the 23 is a very, very,
very large number.

00:01:40.220 --> 00:01:41.240
Where is it useful?

00:01:41.240 --> 00:01:46.280
A mole is useful for counting
atoms, and so what is a mole

00:01:46.280 --> 00:01:49.270
of atoms or molecules?

00:01:49.270 --> 00:01:50.450
What's that many molecules?

00:01:50.450 --> 00:01:54.220
It's 6 followed by roughly 23
0's of molecules-- a very,

00:01:54.220 --> 00:01:55.540
very big number.

00:01:55.540 --> 00:01:57.810
What's interesting about a mole
is that when I have a

00:01:57.810 --> 00:02:05.880
mole of something, its mass--
let's say its mass in grams. A

00:02:05.880 --> 00:02:15.670
mole of carbon: its mass in
grams is going to be equal

00:02:15.670 --> 00:02:22.930
to-- so if I have this many
carbon molecules, its mass in

00:02:22.930 --> 00:02:32.970
grams is x grams. It'll have a
mass of x grams, where x is

00:02:32.970 --> 00:02:43.616
the atomic mass number of an
atom of carbon, although if I

00:02:43.616 --> 00:02:45.750
was talking about a mole of a
molecule, I would figure out

00:02:45.750 --> 00:02:47.770
the atomic mass of the
entire molecule.

00:02:47.770 --> 00:02:49.440
What's an atomic mass number?

00:02:49.440 --> 00:02:54.010
Let me see if I can do a Web
search on a periodic table.

00:02:54.010 --> 00:02:58.800
I'm showing you what I do here,
and it's not fancy.

00:02:58.800 --> 00:03:06.030
Let me go to Google, and let's
look up periodic table, and

00:03:06.030 --> 00:03:10.260
let's see if we can
find a good one--

00:03:10.260 --> 00:03:11.510
this one looks good.

00:03:34.540 --> 00:03:39.410
If we go to carbon, which is
right here, we see that its

00:03:39.410 --> 00:03:41.750
atomic number is 6, and
that's the number of

00:03:41.750 --> 00:03:43.000
protons that it has.

00:03:49.980 --> 00:03:53.500
The atomic mass number-- it's
the mass of the entire atom.

00:03:53.500 --> 00:03:55.890
And just so you know-- we're
delving into a little bit of

00:03:55.890 --> 00:03:59.870
chemistry here-- but most of
the mass of an atom is the

00:03:59.870 --> 00:04:01.410
protons and the neutrons.

00:04:01.410 --> 00:04:05.290
The neutrons and the protons
weigh roughly the same thing,

00:04:05.290 --> 00:04:08.120
and then the electrons are
much, much, much smaller.

00:04:08.120 --> 00:04:12.110
If you factor in the mass of the
protons and the neutrons,

00:04:12.110 --> 00:04:15.090
you pretty much have the
mass of the particle.

00:04:15.090 --> 00:04:18.130
Just a little more chemistry
here is that although on

00:04:18.130 --> 00:04:21.769
average most of the most of
the atoms have roughly the

00:04:21.769 --> 00:04:23.970
same number of protons and
neutrons-- some don't.

00:04:23.970 --> 00:04:28.630
Some, you could have a carbon
atom that has seven neutrons,

00:04:28.630 --> 00:04:30.910
another one with five, another
one with six, and those are

00:04:30.910 --> 00:04:33.260
actually all called isotopes,
and I won't go into all of

00:04:33.260 --> 00:04:35.560
that-- they're just
the same atom with

00:04:35.560 --> 00:04:37.350
different numbers of neutrons.

00:04:37.350 --> 00:04:47.260
In general, the atomic mass is
equal to the mass of the

00:04:47.260 --> 00:04:49.560
protons and the neutrons, and
they tend to be equal.

00:04:49.560 --> 00:04:52.570
So if the atomic number
is 6, the atomic

00:04:52.570 --> 00:04:54.400
mass tends to be 12.

00:04:54.400 --> 00:04:57.220
So why is this useful?

00:04:57.220 --> 00:05:09.390
We can say if we have niobium--
let's say I have a

00:05:09.390 --> 00:05:10.870
mole of niobium.

00:05:10.870 --> 00:05:14.400
If we look here on the periodic
table, it has an

00:05:14.400 --> 00:05:18.840
atomic mass number of 41, and
its average atomic mass-- if

00:05:18.840 --> 00:05:22.770
we were to average all of the
isotopes based on the

00:05:22.770 --> 00:05:25.070
weighting of how they exist
in nature-- it's

00:05:25.070 --> 00:05:27.920
92.9, so roughly 93.

00:05:27.920 --> 00:05:29.860
It's actually a little bit more
than double its atomic

00:05:29.860 --> 00:05:33.600
number, but let's say 93.

00:05:33.600 --> 00:05:39.746
If we had a mole of niobium--
if we had 6.022 times 10 to

00:05:39.746 --> 00:05:48.700
the 23 of niobium, it would
have a mass of 92 grams.

00:05:48.700 --> 00:05:50.560
That's pretty easy-- look
at any element.

00:05:54.570 --> 00:06:00.060
Let's say chromium: we see its
atomic mass number is roughly

00:06:00.060 --> 00:06:02.860
52, and we see that there.

00:06:02.860 --> 00:06:07.530
If I have a mole of it-- if I
have roughly 6 times 10 the 23

00:06:07.530 --> 00:06:12.380
three of it, that much will
have a mass of 52 grams.

00:06:12.380 --> 00:06:13.800
That's how we think
about a mole.

00:06:13.800 --> 00:06:15.930
If I tell you I have a mole of
something, I'm also telling

00:06:15.930 --> 00:06:19.550
you how many of that molecule
I have, and I'm also telling

00:06:19.550 --> 00:06:24.470
you what the mass of that mole
that quantity will be,

00:06:24.470 --> 00:06:25.550
assuming that you
have a periodic

00:06:25.550 --> 00:06:26.720
table in front of you.

00:06:26.720 --> 00:06:30.030
With that said, and with that
out of the way, let's make

00:06:30.030 --> 00:06:32.110
some more progress with
our thermodynamics.

00:06:37.250 --> 00:06:43.990
We said in the last several
videos that pressure times

00:06:43.990 --> 00:06:50.900
volume is somehow
proportional--

00:06:50.900 --> 00:06:51.870
let's call that K.

00:06:51.870 --> 00:06:53.530
And this is an arbitrary number,
it's not some special

00:06:53.530 --> 00:06:57.995
constant-- to the total kinetic
energy of a system.

00:07:01.560 --> 00:07:10.590
We also said that that is
roughly proportional-- that's

00:07:10.590 --> 00:07:18.540
another constant-- times the
number of molecules we have

00:07:18.540 --> 00:07:20.910
times the temperature, because
temperature we viewed as

00:07:20.910 --> 00:07:23.960
kinetic energy per molecule.

00:07:23.960 --> 00:07:27.640
In general, we could also say
that this is proportional to

00:07:27.640 --> 00:07:30.060
this, which is proportional to
this, that pressure times

00:07:30.060 --> 00:07:34.100
volume is proportional.

00:07:34.100 --> 00:07:37.550
We'll use R, because you'll see
where that's coming from

00:07:37.550 --> 00:07:38.730
in a second.

00:07:38.730 --> 00:07:42.160
It's proportional, it's equal
to some constant times the

00:07:42.160 --> 00:07:47.360
number of molecules n.

00:07:47.360 --> 00:07:49.150
Here I just take the absolute
numbers-- if I had five

00:07:49.150 --> 00:07:51.990
molecules, I'd put a five here,
but now this n, I'm

00:07:51.990 --> 00:07:53.240
counting in moles.

00:07:55.670 --> 00:08:00.110
If this n is 1, that means that
I have 6.022 times 10 to

00:08:00.110 --> 00:08:02.490
the 23 molecules.

00:08:02.490 --> 00:08:11.480
One mole equals 6.022 times 10
to the 23, so I'm just saying

00:08:11.480 --> 00:08:13.590
that this is another way to
write the number of molecules,

00:08:13.590 --> 00:08:14.840
and then that's times
temperature.

00:08:18.200 --> 00:08:23.910
Then if we rearrange
it-- PV equals nRT.

00:08:23.910 --> 00:08:26.340
We have a relationship, that
if I know the pressure, the

00:08:26.340 --> 00:08:28.750
volume, and the number of
molecules, I can figure out

00:08:28.750 --> 00:08:31.230
the temperature, or if I know
the number of molecules, the

00:08:31.230 --> 00:08:33.630
temperature, and the pressure,
I can figure out the volume,

00:08:33.630 --> 00:08:37.460
assuming I know what R is.

00:08:37.460 --> 00:08:38.799
I'm about to tell you
what that is.

00:08:38.799 --> 00:08:46.230
R is called the universal gas
constant, and it is R is 8.31

00:08:46.230 --> 00:08:53.030
joules per mole-Kelvin.

00:08:56.210 --> 00:09:00.140
That kind of tells you what
you need in this formula.

00:09:00.140 --> 00:09:05.260
This should end up being
joules, so if you have

00:09:05.260 --> 00:09:08.930
pressure in pascals and volume
in meters cubed, you'll end up

00:09:08.930 --> 00:09:10.050
with joules there.

00:09:10.050 --> 00:09:15.065
This should be in moles--
this is 8.31 joules per

00:09:15.065 --> 00:09:15.740
mole-Kelvin.

00:09:15.740 --> 00:09:20.080
And then this, as we always
said, should be in Kelvin.

00:09:20.080 --> 00:09:24.110
Honestly, if you just memorize
two things in all of

00:09:24.110 --> 00:09:26.090
thermodynamics, you'll probably
be able to do 95% of

00:09:26.090 --> 00:09:27.950
problems, but you actually
should have the intuition of

00:09:27.950 --> 00:09:28.980
how they work.

00:09:28.980 --> 00:09:32.560
Just remember that PV over T
is equal to a constant, or

00:09:32.560 --> 00:09:36.880
that if you change them, they
relate to each other in that

00:09:36.880 --> 00:09:41.860
they all equal a constant, so
P1 times V1 divided by T1 is

00:09:41.860 --> 00:09:44.130
equal to P2 times V2
divided by T2.

00:09:47.310 --> 00:09:52.440
You also should just need to
memorize PV is equal to nRT,

00:09:52.440 --> 00:10:00.000
where R is equal to 8.31
joules per mole Kelvin.

00:10:00.000 --> 00:10:01.660
I know you might not have a
lot of intuition of this

00:10:01.660 --> 00:10:03.580
formula yet, because I haven't
used it, but I'm going to do

00:10:03.580 --> 00:10:04.450
that in the next video.

00:10:04.450 --> 00:10:07.300
These are literally the two
most important things you

00:10:07.300 --> 00:10:09.450
should know in thermodynamics,
and hopefully you have a

00:10:09.450 --> 00:10:11.600
little intuition at this point
of what they mean.

00:10:11.600 --> 00:10:13.030
See you soon.