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PROFESSOR: Today, I'm going to
continue the idea of setting up

00:00:27.110 --> 00:00:29.790
integrals.

00:00:29.790 --> 00:00:39.090
And what we'll deal with
is volumes by slices.

00:00:39.090 --> 00:00:47.240
By slicing.

00:00:47.240 --> 00:00:52.010
And it's lucky that
this is after lunch.

00:00:52.010 --> 00:00:53.960
Maybe it's after
breakfast for some of you,

00:00:53.960 --> 00:00:59.050
because there's the typical way
of introducing this subject is

00:00:59.050 --> 00:01:00.380
with a food analogy.

00:01:00.380 --> 00:01:02.790
There's a lot of ways
of slicing up food.

00:01:02.790 --> 00:01:05.420
And we'll give a few more
examples than just this one.

00:01:05.420 --> 00:01:09.560
But, suppose you
have, well, suppose

00:01:09.560 --> 00:01:13.870
you have a loaf of bread here.

00:01:13.870 --> 00:01:21.077
So here's our loaf
of bread, and I

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hope that looks a little
bit like a loaf of bread.

00:01:23.160 --> 00:01:25.670
It's supposed to be
sitting on the kitchen

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counter ready to be eaten.

00:01:28.920 --> 00:01:32.260
And in order to figure out
how much bread there is there,

00:01:32.260 --> 00:01:35.290
one way of doing it is
to cut it into slices.

00:01:35.290 --> 00:01:37.980
Now, you probably know
that bread is often

00:01:37.980 --> 00:01:40.020
sliced like this.

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There are even
machines to do it.

00:01:42.080 --> 00:01:48.450
And with this setup
here, I'll draw the slice

00:01:48.450 --> 00:01:52.830
with a little bit of a
more colorful decoration.

00:01:52.830 --> 00:01:57.980
So here's our red
slice of bread.

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It's coming around like this.

00:02:00.150 --> 00:02:03.280
And it comes back down behind.

00:02:03.280 --> 00:02:06.960
So here's our bread slice.

00:02:06.960 --> 00:02:21.930
And what I'd like to
figure out is its volume.

00:02:21.930 --> 00:02:24.850
So first of all, there's
the thickness of the bread.

00:02:24.850 --> 00:02:26.950
Which is this
dimension, the thickness

00:02:26.950 --> 00:02:35.490
is this dimension dx here.

00:02:35.490 --> 00:02:37.730
And the only other dimension
that I'm going to give,

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because this is a very
qualitative analysis for now,

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is what I'll call the area.

00:02:47.610 --> 00:02:52.660
And that's the area on
the face of the slice.

00:02:52.660 --> 00:02:57.150
And so the area of
one slice, which

00:02:57.150 --> 00:03:00.240
I'll denote by delta V,
that's a chunk of volume,

00:03:00.240 --> 00:03:05.430
is approximately the area
times the change in x.

00:03:05.430 --> 00:03:13.630
And in the limit, that's going
to be something like this.

00:03:13.630 --> 00:03:16.029
And maybe the areas
of the slices vary.

00:03:16.029 --> 00:03:18.320
There might be a little hole
in the middle of the bread

00:03:18.320 --> 00:03:18.819
somewhere.

00:03:18.819 --> 00:03:21.010
Maybe it gets a little
small on one side.

00:03:21.010 --> 00:03:24.170
So it might change as x changes.

00:03:24.170 --> 00:03:31.760
And the whole volume
you get by adding up.

00:03:31.760 --> 00:03:37.580
So if you like,
this is one slice.

00:03:37.580 --> 00:03:40.830
And this is the sum.

00:03:40.830 --> 00:03:44.810
And you should think of it
in a sort of intuitive way

00:03:44.810 --> 00:03:47.070
as being analogous to the
Riemann sum, where you would

00:03:47.070 --> 00:03:49.390
take each slice individually.

00:03:49.390 --> 00:03:54.300
And that would look like this.

00:03:54.300 --> 00:04:01.720
Alright, so that's just a
superficial and intuitive way

00:04:01.720 --> 00:04:02.830
of looking at it.

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Now, we're only going
to talk about one

00:04:04.810 --> 00:04:06.325
kind of systematic slice.

00:04:06.325 --> 00:04:07.960
It's already on
your problem set,

00:04:07.960 --> 00:04:11.810
you had an example of
a slice of some region.

00:04:11.810 --> 00:04:14.990
But we're only going to talk
systematically about something

00:04:14.990 --> 00:04:32.330
called solids of revolution.

00:04:32.330 --> 00:04:34.430
The idea here is this.

00:04:34.430 --> 00:04:37.930
Suppose you have some
shape, some graph,

00:04:37.930 --> 00:04:40.700
which maybe looks like this.

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And now I'm going to revolve it.

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This is the x-axis and
this is the y-axis.

00:04:46.880 --> 00:04:58.470
In this case, I'm going to
revolve it around the x-axis.

00:04:58.470 --> 00:05:04.760
If you do that, then
the shape that you get

00:05:04.760 --> 00:05:08.310
is maybe like this.

00:05:08.310 --> 00:05:09.930
If I can draw it a little bit.

00:05:09.930 --> 00:05:12.600
It's maybe a football.

00:05:12.600 --> 00:05:17.015
So that's the shape that you get
if you take this piece of disk

00:05:17.015 --> 00:05:18.540
and you revolve it around.

00:05:18.540 --> 00:05:20.500
If you had made this
copy underneath,

00:05:20.500 --> 00:05:22.360
it still would have
been the same region.

00:05:22.360 --> 00:05:27.950
So we only pay attention to
what's above the axis here.

00:05:27.950 --> 00:05:30.600
So that's the basic idea.

00:05:30.600 --> 00:05:33.530
Now, I'm going to apply
the method of slices

00:05:33.530 --> 00:05:35.940
to figure out the
volume of such regions

00:05:35.940 --> 00:05:37.510
and give you a general formula.

00:05:37.510 --> 00:05:40.240
And then apply it
in a specific case.

00:05:40.240 --> 00:05:44.254
I want to take one little
slice of this football,

00:05:44.254 --> 00:05:45.920
maybe a football
wouldn't work too well,

00:05:45.920 --> 00:05:50.380
maybe we should go back
to a loaf of bread.

00:05:50.380 --> 00:05:55.180
Anyway, the key point
is that you never really

00:05:55.180 --> 00:05:58.270
have to draw a 3-D picture.

00:05:58.270 --> 00:05:59.820
And 3-D pictures are awful.

00:05:59.820 --> 00:06:01.600
They're very hard to deal with.

00:06:01.600 --> 00:06:03.460
And it's hard to
visualize with them.

00:06:03.460 --> 00:06:06.010
And one of the reasons why
we're dealing with solids

00:06:06.010 --> 00:06:10.020
of revolution is that we don't
have as many visualization

00:06:10.020 --> 00:06:10.730
problems.

00:06:10.730 --> 00:06:13.170
So we're only going
to deal with this.

00:06:13.170 --> 00:06:15.240
And then you have
to imagine from

00:06:15.240 --> 00:06:17.902
the two-dimensional
cross-section what

00:06:17.902 --> 00:06:19.610
the three-dimensional
picture looks like.

00:06:19.610 --> 00:06:21.372
So we'll do a little
bit of an exercise

00:06:21.372 --> 00:06:22.830
with the
three-dimensional picture.

00:06:22.830 --> 00:06:24.990
But ultimately, you
should be used to,

00:06:24.990 --> 00:06:27.770
getting used to, drawing
2-D diagrams always.

00:06:27.770 --> 00:06:29.710
To depict the
three-dimensional situation.

00:06:29.710 --> 00:06:34.310
Since it's much harder to draw.

00:06:34.310 --> 00:06:38.540
The first step is to consider
what this slice is over here.

00:06:38.540 --> 00:06:45.400
And again it's going
to have width dx.

00:06:45.400 --> 00:06:49.630
And we're going to consider
what it looks like over

00:06:49.630 --> 00:06:51.560
on the 3-D picture.

00:06:51.560 --> 00:06:54.980
So it starts out being
more or less like this.

00:06:54.980 --> 00:06:57.210
But then we're going
to sweep it around.

00:06:57.210 --> 00:06:59.280
We're revolving
around the x-axis,

00:06:59.280 --> 00:07:01.370
so it's spinning
around this way.

00:07:01.370 --> 00:07:03.210
And if you take
this and think of it

00:07:03.210 --> 00:07:06.130
as being on a hinge, which
is down on the x-axis,

00:07:06.130 --> 00:07:09.170
it's going to swing down and
swoop around and come back.

00:07:09.170 --> 00:07:10.750
Swing around.

00:07:10.750 --> 00:07:14.140
And that traces out
something over here.

00:07:14.140 --> 00:07:15.900
Which I'm going
to draw this way.

00:07:15.900 --> 00:07:22.960
It traces out a disk.

00:07:22.960 --> 00:07:29.050
So it's hard for me to draw, and
I'm not going to try too hard.

00:07:29.050 --> 00:07:32.240
Maybe I drew it more like a
wheel, looking like a wheel.

00:07:32.240 --> 00:07:35.850
But anyway, it's this
little flat disk here.

00:07:35.850 --> 00:07:38.110
And so the method
that I'm describing

00:07:38.110 --> 00:07:44.050
for figuring out the volume
is called the method of disks.

00:07:44.050 --> 00:07:56.910
This is going to be
our first method.

00:07:56.910 --> 00:07:58.630
Now I'm going to
apply the reasoning

00:07:58.630 --> 00:08:02.130
that I have up on the
previous blackboard here.

00:08:02.130 --> 00:08:06.180
Namely, I need to get
the volume of this chunk.

00:08:06.180 --> 00:08:08.850
And the way I'm going to
get the volume of this chunk

00:08:08.850 --> 00:08:13.225
is by figuring out its
thickness and its area,

00:08:13.225 --> 00:08:14.880
its cross-sectional area.

00:08:14.880 --> 00:08:19.490
And that's not too
difficult to do.

00:08:19.490 --> 00:08:25.810
If this height, so this height
is what we usually call y.

00:08:25.810 --> 00:08:28.140
And y is usually a function
of x, it's varying.

00:08:28.140 --> 00:08:31.030
And this particular
distance is y.

00:08:31.030 --> 00:08:35.500
Then the area of the face
is easy to calculate.

00:08:35.500 --> 00:08:42.100
Because it's a circle, or a disk
if you like, with this radius.

00:08:42.100 --> 00:08:48.540
So its area is pi y^2.

00:08:48.540 --> 00:08:53.900
So that's, if you like,
one of the dimensions.

00:08:53.900 --> 00:08:59.190
And then the thickness is dx.

00:08:59.190 --> 00:09:03.360
So the incremental
volume is this.

00:09:03.360 --> 00:09:09.140
So this is the method of disks.

00:09:09.140 --> 00:09:11.480
And this is the integrand.

00:09:11.480 --> 00:09:14.410
Now, there's one peculiar
thing about this formula.

00:09:14.410 --> 00:09:17.065
And there are more peculiar
things about this formula.

00:09:17.065 --> 00:09:18.690
But there's one
peculiar thing that you

00:09:18.690 --> 00:09:20.130
should notice immediately.

00:09:20.130 --> 00:09:22.840
Which is that I'm integrating
with respect to x.

00:09:22.840 --> 00:09:26.260
And I haven't yet
told you what y is.

00:09:26.260 --> 00:09:32.280
Well, that will depend on
what function y = f(x) I use.

00:09:32.280 --> 00:09:34.222
So we have to plug
that in eventually.

00:09:34.222 --> 00:09:36.180
If we're actually going
to calculate something,

00:09:36.180 --> 00:09:37.700
we're going to have
to figure that out.

00:09:37.700 --> 00:09:39.070
There's another
very important point

00:09:39.070 --> 00:09:41.340
which is that in order to get
a definite integral, something

00:09:41.340 --> 00:09:42.560
I haven't mentioned,
we're going to have

00:09:42.560 --> 00:09:44.340
to figure out where
we're starting

00:09:44.340 --> 00:09:46.050
and we're ending the picture.

00:09:46.050 --> 00:09:52.250
Which is something we dealt
with last time in 2-D pictures.

00:09:52.250 --> 00:09:55.400
So let's deal with an example.

00:09:55.400 --> 00:10:05.220
And we'll switch over from
a football to a soccer ball.

00:10:05.220 --> 00:10:10.800
I'm going to take a circle,
and we'll say it has radius a.

00:10:10.800 --> 00:10:13.750
So this is 0 and this is a.

00:10:13.750 --> 00:10:17.260
And I'm putting it in this
particular spot for a reason.

00:10:17.260 --> 00:10:19.820
You can do this in
lots of different ways,

00:10:19.820 --> 00:10:22.990
but I'm picking this one
to make a certain exercise

00:10:22.990 --> 00:10:24.970
on your homework easier for you.

00:10:24.970 --> 00:10:31.330
Because I'm doing half
of it for you right now.

00:10:31.330 --> 00:10:32.340
Appreciate it, yeah.

00:10:32.340 --> 00:10:36.180
I'm sure especially
today it's appreciated.

00:10:36.180 --> 00:10:40.330
So again, the formula
has to do with keeping

00:10:40.330 --> 00:10:43.450
track of these slices here.

00:10:43.450 --> 00:10:45.830
And we're sweeping
things around.

00:10:45.830 --> 00:10:48.940
So the full region that
we're talking about

00:10:48.940 --> 00:10:57.300
is the volume of the
ball of radius a.

00:10:57.300 --> 00:10:58.900
That's what our goal
is, to figure out

00:10:58.900 --> 00:11:04.870
what the volume of the
ball of radius a is.

00:11:04.870 --> 00:11:09.670
Alright, again as I say,
the thing sweeps around.

00:11:09.670 --> 00:11:15.800
Coming out of the blackboard,
spinning around on this x-axis.

00:11:15.800 --> 00:11:18.480
So the setup is the following.

00:11:18.480 --> 00:11:21.130
It's always the same.

00:11:21.130 --> 00:11:23.210
Here's our formula.

00:11:23.210 --> 00:11:26.240
And we need to figure out what's
going on with that formula.

00:11:26.240 --> 00:11:32.830
And so we need to solve
for y as a function of x.

00:11:32.830 --> 00:11:36.180
And in order to do that,
what we're going to do

00:11:36.180 --> 00:11:41.890
is just write down the
equation for the circle.

00:11:41.890 --> 00:11:45.800
This is the circle.

00:11:45.800 --> 00:11:49.180
It's centered at (a, 0),
so that's its formula.

00:11:49.180 --> 00:11:51.002
And now there's one
nice thing, which

00:11:51.002 --> 00:11:53.460
is that we really didn't need
it to find the formula for y,

00:11:53.460 --> 00:11:57.140
we only needed to find
the formula for y squared.

00:11:57.140 --> 00:12:00.240
So let's just solve for y^2
and we won't have to solve

00:12:00.240 --> 00:12:01.820
a quadratic or anything.

00:12:01.820 --> 00:12:06.260
Take a square root, that's nice.

00:12:06.260 --> 00:12:12.580
This is a^2 - (x^2 - 2ax + a^2).

00:12:12.580 --> 00:12:15.170
The a^2's cancel.

00:12:15.170 --> 00:12:18.720
These two terms cancel.

00:12:18.720 --> 00:12:30.330
And so the formula
here is 2ax - x^2.

00:12:30.330 --> 00:12:34.670
Alright now, that is what's
known as the integrand.

00:12:34.670 --> 00:12:38.890
Well, except for this
factor of pi here.

00:12:38.890 --> 00:12:42.900
And so the answer for the volume
is going to be the integral

00:12:42.900 --> 00:12:51.050
of pi times this
integrand, (2ax - x^2) dx.

00:12:51.050 --> 00:12:54.450
And that's just the
same thing as this.

00:12:54.450 --> 00:12:57.536
But now there's also
the issue of the limits.

00:12:57.536 --> 00:12:59.160
Which is a completely
separate problem,

00:12:59.160 --> 00:13:03.720
which we also have to solve.

00:13:03.720 --> 00:13:08.650
The range of x is, from
this leftmost point

00:13:08.650 --> 00:13:10.130
to the rightmost point.

00:13:10.130 --> 00:13:13.970
So x varies, starts at 0,
and it goes all the way up

00:13:13.970 --> 00:13:16.310
to what, what's
the top value here.

00:13:16.310 --> 00:13:18.340
2a.

00:13:18.340 --> 00:13:23.290
And so now I have a
completely specified integral.

00:13:23.290 --> 00:13:25.960
Again, and this was
the theme last time,

00:13:25.960 --> 00:13:27.540
the whole goal is
to get ourselves

00:13:27.540 --> 00:13:30.390
to a complete
formula for something

00:13:30.390 --> 00:13:32.021
with an integrand and limits.

00:13:32.021 --> 00:13:33.520
And then we'll be
able to calculate.

00:13:33.520 --> 00:13:40.480
Now, we have clear sailing
to the end of the problem.

00:13:40.480 --> 00:13:42.110
So let's just finish it off.

00:13:42.110 --> 00:13:45.910
We have the volume is, if I
take the antiderivative of that,

00:13:45.910 --> 00:13:53.850
that's pi ax^2, whose derivative
is 2ax, minus x^3 / 3.

00:13:53.850 --> 00:13:57.250
That's the thing whose
derivative is -x^2.

00:13:57.250 --> 00:14:02.290
Evaluated at 0 and 2a.

00:14:02.290 --> 00:14:09.120
And that is equal to
pi times, let's see.

00:14:09.120 --> 00:14:09.930
2^2 = 4.

00:14:09.930 --> 00:14:19.130
So this is 4a^3 -
8a^3 / 3, right?

00:14:19.130 --> 00:14:28.000
And so all told, that is,
let's see, (12/3 - 8/3) pi a^3.

00:14:28.000 --> 00:14:36.670
Which is maybe a familiar
formula, 4/3 pi a^3.

00:14:36.670 --> 00:14:41.540
So it worked, we got it right.

00:14:41.540 --> 00:14:48.060
Let me just point out a
couple of other things

00:14:48.060 --> 00:14:51.020
about this formula.

00:14:51.020 --> 00:14:57.740
The first one is that
from this point of view,

00:14:57.740 --> 00:15:01.910
we've actually accomplished
more then just finding

00:15:01.910 --> 00:15:04.490
the volume of the ball.

00:15:04.490 --> 00:15:07.270
We've also found the
volume of a bunch

00:15:07.270 --> 00:15:11.520
of intermediate regions,
which I can draw schematically

00:15:11.520 --> 00:15:12.690
this way.

00:15:12.690 --> 00:15:18.780
If I chop this thing, and
this portion is x here,

00:15:18.780 --> 00:15:23.160
then the antiderivative
here, this region here,

00:15:23.160 --> 00:15:26.970
which maybe I'll fill
in with this region

00:15:26.970 --> 00:15:31.970
here, which I'm
going to call V(x),

00:15:31.970 --> 00:15:35.500
is the volume of the
portion of the sphere.

00:15:35.500 --> 00:15:53.160
Volume of portion of
width x of the ball.

00:15:53.160 --> 00:16:02.010
And, well, the formula for it is
that it's the volume equals pi

00:16:02.010 --> 00:16:05.530
(ax ^2 - x^3 / 3).

00:16:05.530 --> 00:16:06.654
That's it.

00:16:06.654 --> 00:16:08.320
So we've got something
which is actually

00:16:08.320 --> 00:16:09.600
a lot more information.

00:16:09.600 --> 00:16:16.140
For instance, if
you plug in x = a,

00:16:16.140 --> 00:16:18.500
not surprisingly, and
this is a good idea

00:16:18.500 --> 00:16:20.940
to do because it checks
that we've actually

00:16:20.940 --> 00:16:23.020
got a correct formula here.

00:16:23.020 --> 00:16:28.480
So if you like you can
call this a double-check.

00:16:28.480 --> 00:16:36.220
If you check V(a), this should
be the volume of a half-ball.

00:16:36.220 --> 00:16:38.250
That's halfway.

00:16:38.250 --> 00:16:40.660
If I go over here and
I only go up to a,

00:16:40.660 --> 00:16:42.640
that's exactly half of the ball.

00:16:42.640 --> 00:16:45.400
That had better be
half, so let's just see.

00:16:45.400 --> 00:16:53.830
V(a) in this case is pi, and
then I have (a^3 - a^3 / 3).

00:16:53.830 --> 00:17:00.330
And that turns out to be pi
times a total of 2/3 a^3,

00:17:00.330 --> 00:17:07.540
which is indeed half.

00:17:07.540 --> 00:17:12.490
Now, on your problem set, what
you're going to want to look at

00:17:12.490 --> 00:17:17.870
is this full formula here.

00:17:17.870 --> 00:17:21.420
Of this chunk.

00:17:21.420 --> 00:17:27.140
And what it's going to be good
for is a real-life problem.

00:17:27.140 --> 00:17:30.510
That is, a problem
that really came up

00:17:30.510 --> 00:17:35.930
over the summer, and this fall,
at a couple of universities

00:17:35.930 --> 00:17:38.050
near here, where
people were trying

00:17:38.050 --> 00:17:42.610
to figure out a phenomenon
which is well-known.

00:17:42.610 --> 00:17:56.680
Namely, if you have a bunch
of particles in a fluid,

00:17:56.680 --> 00:18:03.310
and maybe the size of
these things is 1 micron.

00:18:03.310 --> 00:18:06.260
That is, the radius is 1 micron.

00:18:06.260 --> 00:18:09.200
And then you have a bunch
of other little particles,

00:18:09.200 --> 00:18:11.330
which are a lot smaller.

00:18:11.330 --> 00:18:15.750
Maybe 10 nanometers.

00:18:15.750 --> 00:18:23.470
Then what happens is that the
particles, the big particles,

00:18:23.470 --> 00:18:25.030
like to hug each other.

00:18:25.030 --> 00:18:28.120
They like to clump
together, they're very nice.

00:18:28.120 --> 00:18:29.990
Friendly characters.

00:18:29.990 --> 00:18:34.250
So what's the
explanation for this?

00:18:34.250 --> 00:18:38.340
The explanation is
that actually they

00:18:38.340 --> 00:18:40.842
are not quite as friendly
as they might seem.

00:18:40.842 --> 00:18:42.800
What's really happening
is that the little guys

00:18:42.800 --> 00:18:45.240
are shoving them around.

00:18:45.240 --> 00:18:47.380
And pushing them together.

00:18:47.380 --> 00:18:49.700
And they have sharp
elbows, the little ones

00:18:49.700 --> 00:18:50.830
and they're pushing them.

00:18:50.830 --> 00:18:55.480
Don't like them to be around and
they're pushing them together.

00:18:55.480 --> 00:18:58.910
But there's actually
another possibility.

00:18:58.910 --> 00:19:01.550
Which is that they
also will stick

00:19:01.550 --> 00:19:06.180
to the sides of the container.

00:19:06.180 --> 00:19:09.250
So there are two things
that actually happen here.

00:19:09.250 --> 00:19:12.790
And if you want to get
a quantitative handle

00:19:12.790 --> 00:19:16.340
on how much of this
happens, it has

00:19:16.340 --> 00:19:19.500
to do with how much space
these things take up.

00:19:19.500 --> 00:19:22.390
And so the issue is
some kind of overlap

00:19:22.390 --> 00:19:26.000
between a band around
one sphere and a band

00:19:26.000 --> 00:19:27.400
around the other sphere.

00:19:27.400 --> 00:19:32.780
And this overlap region is
what you have to calculate.

00:19:32.780 --> 00:19:36.600
You have to calculate
what's in here.

00:19:36.600 --> 00:19:41.380
And you can do that
using this formula here.

00:19:41.380 --> 00:19:46.300
It's not even difficult. So
this is, if you cut it in half,

00:19:46.300 --> 00:19:48.290
turns out to be
two of these guys.

00:19:48.290 --> 00:19:51.630
And then you're on your way
to figuring out this problem.

00:19:51.630 --> 00:19:53.770
And the question is,
which do they prefer.

00:19:53.770 --> 00:19:55.300
Do they prefer to
touch each other,

00:19:55.300 --> 00:19:57.250
or do they prefer
to touch the wall.

00:19:57.250 --> 00:19:59.050
Do they all cluster to the wall.

00:19:59.050 --> 00:20:02.030
So you can actually see this
in solutions, what they do.

00:20:02.030 --> 00:20:04.020
And the question
is, to what extent

00:20:04.020 --> 00:20:07.580
do they prefer one
configuration to the other.

00:20:07.580 --> 00:20:11.600
So that's a real, live
problem, which really comes up.

00:20:11.600 --> 00:20:13.500
Came up just this year.

00:20:13.500 --> 00:20:15.420
And frequently comes up.

00:20:15.420 --> 00:20:23.640
Which is solved by
our first calculation.

00:20:23.640 --> 00:20:27.350
So now I have, I
called this Method 1.

00:20:27.350 --> 00:20:30.970
For solids of revolution, which
is called the method of disks.

00:20:30.970 --> 00:20:38.670
And now I need to tell you
about the other standard method.

00:20:38.670 --> 00:21:07.850
Which is called the
method of shells.

00:21:07.850 --> 00:21:12.350
So this is our second method.

00:21:12.350 --> 00:21:18.260
I'm going to illustrate this one
with a holiday-themed example

00:21:18.260 --> 00:21:19.740
here.

00:21:19.740 --> 00:21:23.310
This is supposed to be
a witches' cauldron.

00:21:23.310 --> 00:21:28.280
Whoops, witch, witches, well.

00:21:28.280 --> 00:21:33.970
Maybe more than one witch
will have this cauldron here.

00:21:33.970 --> 00:21:35.560
So here's this shape.

00:21:35.560 --> 00:21:44.500
And we're going to figure out
how much liquid is in here.

00:21:44.500 --> 00:21:47.760
I'm going to plot this.

00:21:47.760 --> 00:21:54.210
Maybe I'll put this down
just a bit lower here.

00:21:54.210 --> 00:21:58.750
And I'm going to
make it a parabola.

00:21:58.750 --> 00:22:01.510
This is y = x^2.

00:22:01.510 --> 00:22:08.940
And I'm going to make
the top height be y = a.

00:22:08.940 --> 00:22:10.260
So here's my situation.

00:22:10.260 --> 00:22:14.750
And I want to figure out
how much liquid is in here.

00:22:14.750 --> 00:22:18.050
Now, the reason why I presented
the problem in this form,

00:22:18.050 --> 00:22:20.780
of course, is really to get
you used to these things.

00:22:20.780 --> 00:22:23.490
And the first new thing that
I want you to get used to

00:22:23.490 --> 00:22:28.160
is the idea that now we can also
revolve around the y-axis, not

00:22:28.160 --> 00:22:30.580
just the x-axis.

00:22:30.580 --> 00:22:40.550
So this one is going to be
revolved around the y-axis.

00:22:40.550 --> 00:22:41.550
And that's what happens.

00:22:41.550 --> 00:22:43.790
If you spin the
parabola around, you

00:22:43.790 --> 00:22:54.030
get this kind of shape, this
kind of solid shape here.

00:22:54.030 --> 00:22:59.280
Now, I'm going to
use the same kind

00:22:59.280 --> 00:23:01.530
of slicing that I did before.

00:23:01.530 --> 00:23:04.370
But it's going to look
totally different.

00:23:04.370 --> 00:23:09.180
Namely, I'll draw
it in red again.

00:23:09.180 --> 00:23:12.820
I'm going to take a
little slice over here.

00:23:12.820 --> 00:23:16.580
And now I want to imagine
what happens if it gets

00:23:16.580 --> 00:23:23.060
revolved around the y-axis.

00:23:23.060 --> 00:23:24.699
This time it's not a disk.

00:23:24.699 --> 00:23:26.740
Actually, if I revolve
this around the other way,

00:23:26.740 --> 00:23:28.140
it would have had a hole in it.

00:23:28.140 --> 00:23:30.410
Which is also possible to do.

00:23:30.410 --> 00:23:34.050
That's practically the same
as the method of disks.

00:23:34.050 --> 00:23:38.387
You'll maybe discuss
that in recitation.

00:23:38.387 --> 00:23:40.470
Anyway, we're going to
revolve it around this way.

00:23:40.470 --> 00:23:43.760
So again, I need
to sweep it around,

00:23:43.760 --> 00:23:47.080
swing it around like this.

00:23:47.080 --> 00:23:54.590
And I'll draw the shape.

00:23:54.590 --> 00:23:56.180
It's going to sweep
around in a circle

00:23:56.180 --> 00:24:00.300
and maybe it'll have a little
bit of thickness to it.

00:24:00.300 --> 00:24:06.620
And this is the thing
that people call a shell.

00:24:06.620 --> 00:24:12.360
This is the so-called
shell of the method.

00:24:12.360 --> 00:24:14.900
I would maybe call
it a cylinder,

00:24:14.900 --> 00:24:16.690
and another way of
thinking about it

00:24:16.690 --> 00:24:22.350
is that you can maybe
wrap up a piece of paper.

00:24:22.350 --> 00:24:27.310
Like this.

00:24:27.310 --> 00:24:29.440
There it is, there's a
cylinder you can see.

00:24:29.440 --> 00:24:31.440
Very thin, right?

00:24:31.440 --> 00:24:34.990
Very thin.

00:24:34.990 --> 00:24:37.590
Now, the reason why I
used the piece of paper

00:24:37.590 --> 00:24:39.870
as an example of
this is that I'm

00:24:39.870 --> 00:24:43.220
going to have to figure out
the volume of this thing.

00:24:43.220 --> 00:24:56.110
Its thickness, as
usual, is equal to dx.

00:24:56.110 --> 00:25:01.710
Its height is what?

00:25:01.710 --> 00:25:05.570
Well, actually I have to
use the diagram to see that.

00:25:05.570 --> 00:25:15.340
The top value is a, and the
bottom value is what we call y.

00:25:15.340 --> 00:25:22.210
So the height is equal
to, I'm sorry, a - y.

00:25:22.210 --> 00:25:25.127
Now, again this is an
incredibly risky thing here.

00:25:25.127 --> 00:25:26.960
And I've done this
before I pointed this out

00:25:26.960 --> 00:25:29.340
on the very first
day of lecture.

00:25:29.340 --> 00:25:34.040
The letter y represents a
lot of different things.

00:25:34.040 --> 00:25:39.460
And in disguise, when I
call this y, I mean y = x^2.

00:25:39.460 --> 00:25:41.200
In other words, the
interesting curve.

00:25:41.200 --> 00:25:45.080
I don't mean y = a,
which is the other part.

00:25:45.080 --> 00:25:55.080
In general, you might think of
it as being equal to y_(top) -

00:25:55.080 --> 00:25:57.780
y_(bottom).

00:25:57.780 --> 00:26:00.140
So there are, of course,
two y's involved.

00:26:00.140 --> 00:26:01.390
In disguise.

00:26:01.390 --> 00:26:04.450
But we have a
shorthand, and the sort

00:26:04.450 --> 00:26:06.290
of uninteresting
one we call by its--

00:26:06.290 --> 00:26:09.340
we just evaluate immediately,
and the interesting one

00:26:09.340 --> 00:26:14.330
we leave as the symbol y.

00:26:14.330 --> 00:26:18.030
Now, the last bit that
I have to do, I claim,

00:26:18.030 --> 00:26:22.330
in order to figure this
out, is the circumference.

00:26:22.330 --> 00:26:26.130
And the reason for that is
that if I think of this thing

00:26:26.130 --> 00:26:30.260
as like this tube,
or piece of paper

00:26:30.260 --> 00:26:33.500
here, in order to figure out
how much stuff there is here,

00:26:33.500 --> 00:26:39.160
all I have to do is unfold it.

00:26:39.160 --> 00:26:42.629
Its size, the whole
quantity of paper

00:26:42.629 --> 00:26:44.670
here is the same whether
it's rolled up like this

00:26:44.670 --> 00:26:46.090
or whether it's stretched out.

00:26:46.090 --> 00:26:56.110
So if I unwrap it,
it looks like what?

00:26:56.110 --> 00:26:58.790
Well, it looks like
kind of a slab?

00:26:58.790 --> 00:27:06.550
Right, it looks like
just a slab like this.

00:27:06.550 --> 00:27:08.860
And again, the thickness is dx.

00:27:08.860 --> 00:27:10.650
The height is a - y.

00:27:10.650 --> 00:27:13.990
And now we can see
that the length here

00:27:13.990 --> 00:27:16.100
is all the way around.

00:27:16.100 --> 00:27:19.710
It's the circumference.

00:27:19.710 --> 00:27:24.100
So this is going to be the
circumference when I unwrap it.

00:27:24.100 --> 00:27:26.520
And in order to figure
out the circumference,

00:27:26.520 --> 00:27:29.510
I need to figure out the radius.

00:27:29.510 --> 00:27:34.800
So the radius is, on this
diagram, is right over there.

00:27:34.800 --> 00:27:38.160
This is the radius.

00:27:38.160 --> 00:27:42.430
And that distance is x.

00:27:42.430 --> 00:27:45.850
So this length here is x.

00:27:45.850 --> 00:27:56.440
And so this
circumference is 2 pi x.

00:27:56.440 --> 00:28:00.520
And this height is
still ay. a - y, sorry.

00:28:00.520 --> 00:28:13.610
And the thickness is still dx.

00:28:13.610 --> 00:28:17.550
So in total, we're just going to
multiply these numbers together

00:28:17.550 --> 00:28:27.110
to get the total volume.

00:28:27.110 --> 00:28:31.260
We have, in other words,
dV is equal to the product

00:28:31.260 --> 00:28:37.040
of the (2 pi x) dimension
dimension, the (8 - y)

00:28:37.040 --> 00:28:42.150
dimension, and the dx dimension.

00:28:42.150 --> 00:28:43.790
Incidentally,
dimensional analysis

00:28:43.790 --> 00:28:46.700
is very useful and
important in these problems.

00:28:46.700 --> 00:28:49.167
You can see that there
are three lengths being

00:28:49.167 --> 00:28:50.000
multiplied together.

00:28:50.000 --> 00:28:52.140
So we'll get a volume in end.

00:28:52.140 --> 00:28:52.840
Something cubic.

00:28:52.840 --> 00:28:54.839
And we will be coming
back to that, because it's

00:28:54.839 --> 00:28:57.690
a quite subtle issue sometimes.

00:28:57.690 --> 00:29:01.720
So here's the formula, and
let's simplify it a little bit.

00:29:01.720 --> 00:29:08.690
We have 2 pi x times, remember,
first I have to substitute,

00:29:08.690 --> 00:29:11.450
otherwise I'm never going
to be able to integrate.

00:29:11.450 --> 00:29:22.460
And then I rewrite that as 2
pi (ax - x^2, whoops, x^3) dx.

00:29:22.460 --> 00:29:25.470
Better not get that wrong.

00:29:25.470 --> 00:29:29.280
And now the last little
bit here, that I had better

00:29:29.280 --> 00:29:31.570
be careful about in
order to figure out

00:29:31.570 --> 00:29:37.900
what the total volume
is, is the limits.

00:29:37.900 --> 00:29:42.020
So the volume is going to be the
integral of this quantity 2 pi

00:29:42.020 --> 00:29:48.600
(ax - x^3) dx.

00:29:48.600 --> 00:29:58.660
And now I have to pay attention
to what the limits are.

00:29:58.660 --> 00:30:02.130
Now, here you have
to be careful.

00:30:02.130 --> 00:30:06.770
x is possibly the-- You
have to always go back

00:30:06.770 --> 00:30:08.200
to the 2-D diagram.

00:30:08.200 --> 00:30:10.810
I went to it immediately,
but that's the whole point.

00:30:10.810 --> 00:30:14.990
Is that everything gets read
off from this diagram here.

00:30:14.990 --> 00:30:17.380
When you take this guy
and you sweep it around,

00:30:17.380 --> 00:30:20.330
you take care of everything
that's to the left.

00:30:20.330 --> 00:30:22.830
So we only have to count
what's to the right.

00:30:22.830 --> 00:30:24.580
We don't have to count
anything over here.

00:30:24.580 --> 00:30:27.740
Because it's taken care
of when we sweep around.

00:30:27.740 --> 00:30:31.490
So the starting place
is going to be x = 0.

00:30:31.490 --> 00:30:34.810
That's where we start.

00:30:34.810 --> 00:30:41.070
And where we end is the
farthest, rightmost spot for x.

00:30:41.070 --> 00:30:43.680
Which is down here.

00:30:43.680 --> 00:30:45.930
You've got to watch out
about where that is.

00:30:45.930 --> 00:30:48.300
In the y variable,
it's up at y = a.

00:30:48.300 --> 00:30:53.080
But in the x variable, we
can see that it's what?

00:30:53.080 --> 00:30:56.230
It's the square root of a.

00:30:56.230 --> 00:30:58.700
So these limits,
this is where you're

00:30:58.700 --> 00:31:01.767
going to focus all your
attention on the integrand

00:31:01.767 --> 00:31:02.850
and getting it just right.

00:31:02.850 --> 00:31:04.790
And then you're going
to lose your steam

00:31:04.790 --> 00:31:06.830
and not pay attention
to the limits.

00:31:06.830 --> 00:31:09.619
They're equally important.

00:31:09.619 --> 00:31:12.035
You've no hope of getting the
right answer without getting

00:31:12.035 --> 00:31:14.060
the limits right.

00:31:14.060 --> 00:31:17.020
So this is the integral
from 0 to square root of a.

00:31:17.020 --> 00:31:26.463
Again, that's just because y =
a and y = x^2 implies x = square

00:31:26.463 --> 00:31:27.950
root a.

00:31:27.950 --> 00:31:30.300
That's that upper limit there.

00:31:30.300 --> 00:31:33.900
And now, we're ready
to carry out this,

00:31:33.900 --> 00:31:35.490
to evaluate this integral.

00:31:35.490 --> 00:31:39.650
So we get 2-- sorry, we get
2 pi ax, that's pi ax^2.

00:31:42.680 --> 00:31:47.060
Maybe I'll leave
the 2's in there.

00:31:47.060 --> 00:31:52.210
2 pi ax^2 is the antiderivative
of this ax^2 / 2,

00:31:52.210 --> 00:31:59.810
and then here x^4 / 4, evaluated
at 0 and square root of a.

00:31:59.810 --> 00:32:05.560
And finally, let's
see, what is that.

00:32:05.560 --> 00:32:14.410
That's 2 pi (a^2 / 2 - a^2 / 4).

00:32:14.410 --> 00:32:19.900
Which is a total of 1/4,
right, 2 pi a^2 / 4.

00:32:19.900 --> 00:32:23.990
Which is pi/2 a^2.

00:32:33.210 --> 00:32:33.920
Yes, question.

00:32:33.920 --> 00:32:40.322
STUDENT: [INAUDIBLE]

00:32:40.322 --> 00:32:41.030
PROFESSOR: Right.

00:32:41.030 --> 00:32:43.570
So the question is,
why did I integrate

00:32:43.570 --> 00:32:46.890
only from the
middle to this end,

00:32:46.890 --> 00:32:48.560
instead of all the
way from over here,

00:32:48.560 --> 00:32:52.150
minus square root of a, all
the way to the plus end.

00:32:52.150 --> 00:32:55.220
And the reason is that
you have to look at what's

00:32:55.220 --> 00:32:57.470
happening with the rotation.

00:32:57.470 --> 00:33:00.820
This red guy, when
I swept it around,

00:33:00.820 --> 00:33:04.350
I counted the stuff to
the right and the left.

00:33:04.350 --> 00:33:09.680
So in other words, if I just
rotate the right half of this,

00:33:09.680 --> 00:33:12.210
I'm covering the left half.

00:33:12.210 --> 00:33:15.970
So if I counted the stuff
from minus square root of a,

00:33:15.970 --> 00:33:17.190
I would be doing it twice.

00:33:17.190 --> 00:33:19.910
I would be doubling what I need.

00:33:19.910 --> 00:33:21.040
So it's too much.

00:33:21.040 --> 00:33:23.450
Another way of saying
it is if I wanted

00:33:23.450 --> 00:33:25.810
to take the whole
region, if I rotate it

00:33:25.810 --> 00:33:29.380
around only 180
degrees, only pi, that

00:33:29.380 --> 00:33:32.610
would fill up the whole
region, if I did both halves.

00:33:32.610 --> 00:33:35.000
And then instead of a
circumference, instead of a 2

00:33:35.000 --> 00:33:36.850
pi x, I could use a pi x.

00:33:36.850 --> 00:33:39.180
But then I would have
double what I had.

00:33:39.180 --> 00:33:41.980
So there are two ways
of looking at it.

00:33:41.980 --> 00:33:45.140
The same goes, actually,
for the football case.

00:33:45.140 --> 00:33:47.930
When I have that football, I
didn't count the bottom part.

00:33:47.930 --> 00:33:50.940
Because when I swung
it around the x-axis,

00:33:50.940 --> 00:33:57.870
the top part sufficed and I
could ignore the bottom half.

00:33:57.870 --> 00:33:59.140
Yeah, another question.

00:33:59.140 --> 00:34:04.010
STUDENT: [INAUDIBLE]

00:34:04.010 --> 00:34:05.260
PROFESSOR: Ooh, good question.

00:34:05.260 --> 00:34:08.290
The question is, when
do you know, how do you

00:34:08.290 --> 00:34:12.910
know when to take the rectangle
to be vertical or horizontal.

00:34:12.910 --> 00:34:15.850
So far we've only done
vertical rectangles.

00:34:15.850 --> 00:34:18.350
And I'm going to do a
horizontal example in a second.

00:34:18.350 --> 00:34:23.560
And the answer to the question
of when you do it is this.

00:34:23.560 --> 00:34:26.300
You can always set
it up both ways.

00:34:26.300 --> 00:34:29.560
One way may be a
difficult calculation

00:34:29.560 --> 00:34:33.130
and one way may be an
easier calculation.

00:34:33.130 --> 00:34:36.079
Yesterday, we did it -
or, sorry, the last time.

00:34:36.079 --> 00:34:37.370
Yeah, I guess it was yesterday.

00:34:37.370 --> 00:34:39.890
We did it with-- and the
horizontal and the vertical

00:34:39.890 --> 00:34:41.954
were quite different
in character.

00:34:41.954 --> 00:34:43.870
One of them was really
a mess, and one of them

00:34:43.870 --> 00:34:45.130
was a little easier.

00:34:45.130 --> 00:34:47.620
So very often, one will
be easier than the other.

00:34:47.620 --> 00:34:50.790
Every once in a while,
one of them is impossible

00:34:50.790 --> 00:34:52.640
and the other one is possible.

00:34:52.640 --> 00:34:55.210
In other words, the difference
in difficulty can be extreme.

00:34:55.210 --> 00:34:58.670
So you don't know
that in advance.

00:34:58.670 --> 00:34:59.870
Yeah, another question.

00:34:59.870 --> 00:35:00.990
STUDENT: [INAUDIBLE]

00:35:00.990 --> 00:35:02.790
PROFESSOR: The
question is, did we

00:35:02.790 --> 00:35:06.580
just find the volume
when you rotate

00:35:06.580 --> 00:35:11.510
this green region around.

00:35:11.510 --> 00:35:17.290
Or, did we find the volume when
we rotate this whole region.

00:35:17.290 --> 00:35:19.711
In other words, just the right
half or the right and the

00:35:19.711 --> 00:35:20.210
left half.

00:35:20.210 --> 00:35:21.870
The answer is both.

00:35:21.870 --> 00:35:25.280
The region that you
get is the same.

00:35:25.280 --> 00:35:27.114
You always get this
cauldron, whether you

00:35:27.114 --> 00:35:29.030
take this right half
when you rotate it around

00:35:29.030 --> 00:35:32.974
or you take both and
you rotate it around.

00:35:32.974 --> 00:35:35.140
So the answer to both of
those questions is the same

00:35:35.140 --> 00:35:35.730
and it's this.

00:35:35.730 --> 00:35:37.170
Yes.

00:35:37.170 --> 00:35:43.207
STUDENT: [INAUDIBLE]

00:35:43.207 --> 00:35:45.290
PROFESSOR: So that if you
rotated them both around

00:35:45.290 --> 00:35:47.580
and you only wanted
to cover things once,

00:35:47.580 --> 00:35:49.350
you would rotate halfway around.

00:35:49.350 --> 00:35:52.070
Only by 180 degrees.

00:35:52.070 --> 00:35:53.040
That's true.

00:35:53.040 --> 00:35:55.379
But you can rotate around
as many times as you want.

00:35:55.379 --> 00:35:56.920
You're still covering
the same thing.

00:35:56.920 --> 00:36:03.870
Over and over and
over and over again.

00:36:03.870 --> 00:36:05.570
So let's go on.

00:36:05.570 --> 00:36:08.920
I have a very subtle point
that I need to discuss with you

00:36:08.920 --> 00:36:23.640
in order to go on to
the next application.

00:36:23.640 --> 00:36:25.820
So here's my, and
I do want to get

00:36:25.820 --> 00:36:29.190
to the point of horizontal
cross-sections as well.

00:36:29.190 --> 00:36:32.450
So let's continue here.

00:36:32.450 --> 00:36:37.860
So the first thing that I
want to point out to you now

00:36:37.860 --> 00:36:50.230
is, I want you to
beware of units.

00:36:50.230 --> 00:36:54.570
There's something a little
fishy in this problem.

00:36:54.570 --> 00:37:01.660
And it can be summarized in the
character of the answer, which

00:37:01.660 --> 00:37:05.320
is just a little bit not clear.

00:37:05.320 --> 00:37:09.860
Namely, it looks
like it's a^2, right?

00:37:09.860 --> 00:37:12.710
And we know that a
is in units and we

00:37:12.710 --> 00:37:14.950
should've gotten cubic units.

00:37:14.950 --> 00:37:18.520
So there's something a little
bit tricky about this question.

00:37:18.520 --> 00:37:22.190
And so I want to illustrate
the paradox right now.

00:37:22.190 --> 00:37:26.050
So, suppose that a
= 100 centimeters.

00:37:26.050 --> 00:37:29.080
And suppose the units
are centimeters.

00:37:29.080 --> 00:37:33.610
Then the formula for the
volume is pi/2 100^2.

00:37:36.640 --> 00:37:40.336
and the units we must take
are centimeters cubed.

00:37:40.336 --> 00:37:41.710
Despite the fact
that you kind of

00:37:41.710 --> 00:37:43.290
want to square the centimeters.

00:37:43.290 --> 00:37:45.740
But that's not what
this problem says.

00:37:45.740 --> 00:37:50.200
OK, so this is the
situation that we've got.

00:37:50.200 --> 00:37:54.750
Now, if you work
out what this is,

00:37:54.750 --> 00:37:58.710
to figure out what the
volume of this cauldron is,

00:37:58.710 --> 00:38:04.010
what you find is that
it's pi / 2 times, well,

00:38:04.010 --> 00:38:10.160
10^4 is 10 * 1,000
centimeters cubed.

00:38:10.160 --> 00:38:12.410
And those are otherwise
known as liters.

00:38:12.410 --> 00:38:15.560
So this is approximately
10 pi / 2 liters,

00:38:15.560 --> 00:38:20.710
which is about 16 liters.

00:38:20.710 --> 00:38:23.230
And so that's how much
was in the cauldron

00:38:23.230 --> 00:38:27.190
under this choice of units.

00:38:27.190 --> 00:38:30.010
Now, I'm going to make
another choice of units now.

00:38:30.010 --> 00:38:32.420
And we're going to
make a comparison.

00:38:32.420 --> 00:38:38.510
Suppose that the
units are 1 meter.

00:38:38.510 --> 00:38:40.370
Looks like it
should be the same,

00:38:40.370 --> 00:38:47.260
but if I calculate the volume,
it's going to be pi / 2,

00:38:47.260 --> 00:38:51.610
and 1^2 times meters cubed.

00:38:51.610 --> 00:38:55.600
That's what the
formula tells us to do.

00:38:55.600 --> 00:39:04.860
And if you calculate that
out, that's pi / 2 (100 cm)^3.

00:39:04.860 --> 00:39:09.660
And with this unit
notation we really do

00:39:09.660 --> 00:39:12.240
want to cube the centimeters
and cube the 100.

00:39:12.240 --> 00:39:14.880
So we get pi / 2.

00:39:14.880 --> 00:39:18.540
And here we get 10 ^ 6 cm^3.

00:39:18.540 --> 00:39:26.990
And that comes out to
pi / 2 * 1000 liters.

00:39:26.990 --> 00:39:35.560
Or, in other words,
about 1600 liters.

00:39:35.560 --> 00:39:40.420
So I'd like to ask you
first to contemplate this.

00:39:40.420 --> 00:39:42.490
And this is a paradox.

00:39:42.490 --> 00:39:44.360
And this is a serious paradox.

00:39:44.360 --> 00:39:45.910
If you really want
to apply problems,

00:39:45.910 --> 00:39:49.650
you actually have to understand
what your answers mean.

00:39:49.650 --> 00:39:53.411
So what do you think
is going on here?

00:39:53.411 --> 00:39:53.910
Yeah.

00:39:53.910 --> 00:40:06.916
STUDENT: [INAUDIBLE]

00:40:06.916 --> 00:40:07.540
PROFESSOR: Yes.

00:40:07.540 --> 00:40:19.252
STUDENT: [INAUDIBLE]

00:40:19.252 --> 00:40:19.960
PROFESSOR: Right.

00:40:19.960 --> 00:40:22.980
So the question is, how could
either of these make sense.

00:40:22.980 --> 00:40:26.700
How am I dealing with
the units in either case.

00:40:26.700 --> 00:40:30.700
So now I'm going to
explain to you the answer.

00:40:30.700 --> 00:40:32.590
Because this is
really quite puzzling.

00:40:32.590 --> 00:40:36.470
But it has a resolution.

00:40:36.470 --> 00:40:49.220
The answer to this question is
that both answers are correct.

00:40:49.220 --> 00:40:54.050
This is correct
reasoning in both cases.

00:40:54.050 --> 00:41:00.080
What's the matter is that you
have to interpret the equation

00:41:00.080 --> 00:41:13.800
y = x^2 in two different ways.

00:41:13.800 --> 00:41:17.060
Two ways.

00:41:17.060 --> 00:41:19.650
So let me explain what they are.

00:41:19.650 --> 00:41:31.950
For instance, you can take
y = x^2 in centimeters.

00:41:31.950 --> 00:41:35.610
So y = x ^2 in centimeters.

00:41:35.610 --> 00:41:44.850
In which case, the picture
looks like the following.

00:41:44.850 --> 00:41:49.100
a = 100 centimeters.

00:41:49.100 --> 00:41:55.990
And this distance here, which
is the x value, this is 10.

00:41:55.990 --> 00:41:57.810
This is 10 centimeters.

00:41:57.810 --> 00:41:59.790
And that's what the
relationship means.

00:41:59.790 --> 00:42:02.930
So the top of the
cauldron, if you like,

00:42:02.930 --> 00:42:06.879
this distance here
is 20 centimeters.

00:42:06.879 --> 00:42:08.670
This is actually very
badly drawn to scale.

00:42:08.670 --> 00:42:10.890
It's actually very,
very, deep, this thing.

00:42:10.890 --> 00:42:13.900
It's a rather skinny, deep one.

00:42:13.900 --> 00:42:19.290
So this is very much not
to scale, this picture.

00:42:19.290 --> 00:42:27.920
The other picture, the other
picture is interpreting y = x^2

00:42:27.920 --> 00:42:34.350
in meters.

00:42:34.350 --> 00:42:37.140
And that's more like what
I had in mind, actually.

00:42:37.140 --> 00:42:40.340
I had in mind this big vat here.

00:42:40.340 --> 00:42:46.530
And this distance
here is 1 meter,

00:42:46.530 --> 00:42:49.440
and then the square
root of 1 is 1.

00:42:49.440 --> 00:42:52.770
So this distance
here is also 1 meter.

00:42:52.770 --> 00:43:00.210
And the top is 2 meters.

00:43:00.210 --> 00:43:02.410
Now, it's not that crazy.

00:43:02.410 --> 00:43:04.410
And in fact it's easy to
check that, it's pretty

00:43:04.410 --> 00:43:06.040
reasonable in terms of scale.

00:43:06.040 --> 00:43:09.600
That this thing has
16 liters in it.

00:43:09.600 --> 00:43:17.390
And this guy has
1600 liters in it.

00:43:17.390 --> 00:43:20.370
So you actually have to
know what your symbols mean

00:43:20.370 --> 00:43:23.120
when you're dealing with these
kinds of applied problems.

00:43:23.120 --> 00:43:27.170
And if you're ever really going
to do some real consequences,

00:43:27.170 --> 00:43:30.260
you have to know
what the units mean.

00:43:30.260 --> 00:43:32.870
And the problem with the
equation y = x^2 is that

00:43:32.870 --> 00:43:37.600
it's the one that
violated scaling rules.

00:43:37.600 --> 00:43:38.100
Yeah.

00:43:38.100 --> 00:43:47.264
STUDENT: [INAUDIBLE]

00:43:47.264 --> 00:43:47.930
PROFESSOR: Yeah.

00:43:47.930 --> 00:43:52.927
STUDENT: [INAUDIBLE]

00:43:52.927 --> 00:43:53.510
PROFESSOR: No.

00:43:53.510 --> 00:43:59.300
STUDENT: [INAUDIBLE]

00:43:59.300 --> 00:44:09.020
PROFESSOR: OK, so the question
is whether the formula V = pi /

00:44:09.020 --> 00:44:11.520
2 a^2.

00:44:11.520 --> 00:44:14.430
This is the correct
answer to the problem.

00:44:14.430 --> 00:44:17.920
But it is not
consistent in units.

00:44:17.920 --> 00:44:21.340
If you plug in a equals
some number of centimeters,

00:44:21.340 --> 00:44:24.060
some number of millimeters,
some number of inches and so on,

00:44:24.060 --> 00:44:26.380
every single time you'll
get a different answer.

00:44:26.380 --> 00:44:30.160
And they're all inconsistent.

00:44:30.160 --> 00:44:33.800
In other words, this
formula violates scaling.

00:44:33.800 --> 00:44:36.960
STUDENT: [INAUDIBLE]

00:44:36.960 --> 00:44:44.240
PROFESSOR: If you study
each step correctly,

00:44:44.240 --> 00:44:47.342
you will discover that these
are the consistent and correct

00:44:47.342 --> 00:44:49.300
statements, what I'm
writing on the blackboard.

00:44:49.300 --> 00:44:53.060
And this makes sense
in a unit-less sense.

00:44:53.060 --> 00:44:56.850
But then if you actually stick
units on them, one of them,

00:44:56.850 --> 00:44:58.200
they both are correct.

00:44:58.200 --> 00:45:00.590
And one of them describes
this situation and one of them

00:45:00.590 --> 00:45:03.850
describes this situation.

00:45:03.850 --> 00:45:07.100
And it's a mistake to think
of this as being 1 meter

00:45:07.100 --> 00:45:09.910
and cubing the meters.

00:45:09.910 --> 00:45:20.420
That will be an error that
will cause you problems.

00:45:20.420 --> 00:45:23.370
This 1 is just unit-less,
and then the meters cubed

00:45:23.370 --> 00:45:30.320
got converted.

00:45:30.320 --> 00:45:33.250
So I encourage you to
study this on your own.

00:45:33.250 --> 00:45:35.770
So now I'm going to
introduce the next problem.

00:45:35.770 --> 00:45:38.580
We'll have to
solve it next time.

00:45:38.580 --> 00:45:41.160
But the reason why I spent
all the time on units

00:45:41.160 --> 00:45:43.300
is that otherwise it
would be impossible

00:45:43.300 --> 00:45:46.934
for you to believe me when
I do this next calculation.

00:45:46.934 --> 00:45:48.350
Because we're
trying to get a real

00:45:48.350 --> 00:45:50.360
answer out of a real question.

00:45:50.360 --> 00:45:53.210
And I'm going to make
conversions between centimeters

00:45:53.210 --> 00:45:55.080
and meters back and forth.

00:45:55.080 --> 00:45:57.080
And we have to get it
consistent in order

00:45:57.080 --> 00:45:58.200
to have the right answer.

00:45:58.200 --> 00:46:04.420
So there was a reason for
illustrating this pitfall.

00:46:04.420 --> 00:46:08.400
So this second, the next
thing that I'd like to do,

00:46:08.400 --> 00:46:19.560
is I'd like to boil the water
in the witches' cauldron.

00:46:19.560 --> 00:46:23.480
This is definitely
seasonally appropriate,

00:46:23.480 --> 00:46:25.190
since we're
approaching Halloween.

00:46:25.190 --> 00:46:29.500
And we'll work it
out fully next time.

00:46:29.500 --> 00:46:35.260
Now, I'm going to
introduce another feature

00:46:35.260 --> 00:46:36.301
into the problem.

00:46:36.301 --> 00:46:38.550
And this is the one that I
want you to understand now.

00:46:38.550 --> 00:46:41.527
We'll set it all up
tomorrow, but right now I

00:46:41.527 --> 00:46:43.860
need you to understand what
the new main idea that we're

00:46:43.860 --> 00:46:51.400
going to get.

00:46:51.400 --> 00:46:53.590
There is the new
physical feature

00:46:53.590 --> 00:46:56.170
that I'm going to add to this
problem is that if when you're

00:46:56.170 --> 00:47:01.150
boiling, when the witches
are boiling this water,

00:47:01.150 --> 00:47:03.070
the temperature of
the water is not

00:47:03.070 --> 00:47:06.617
the same at each
level in the kettle.

00:47:06.617 --> 00:47:08.950
At the bottom of the kettle,
where you're heating it up,

00:47:08.950 --> 00:47:11.150
it's at its highest temperature.

00:47:11.150 --> 00:47:15.291
So at the bottom it's going
to be, say, 100 degrees.

00:47:15.291 --> 00:47:17.040
That is, it's going
to be totally boiling.

00:47:17.040 --> 00:47:18.830
100 degrees Celsius.

00:47:18.830 --> 00:47:30.690
And at the top, it's going
to be, say, 70 degrees.

00:47:30.690 --> 00:47:32.150
Right, it's very cold outside.

00:47:32.150 --> 00:47:37.420
In fact, it's 0 degrees outside.

00:47:37.420 --> 00:47:40.920
Which is the temperature at
which all witches operate,

00:47:40.920 --> 00:47:41.900
I think.

00:47:41.900 --> 00:47:46.160
Anyway, so they're
boiling their stuff.

00:47:46.160 --> 00:47:48.230
And the question that
we're going to ask

00:47:48.230 --> 00:47:55.540
is how much heat, how much
heat, do they need to do it.

00:47:55.540 --> 00:48:00.170
Now, the thing starts
out at 0 degrees Celsius.

00:48:00.170 --> 00:48:02.820
And we're going to heat it up to
this temperature configuration

00:48:02.820 --> 00:48:03.490
here.

00:48:03.490 --> 00:48:07.750
But it's rising from 100
down here to 70 up here.

00:48:07.750 --> 00:48:10.920
So the temperature
is varying in height.

00:48:10.920 --> 00:48:12.660
And for simplicity
I'm going to make

00:48:12.660 --> 00:48:19.700
the formula for the temperature
be 70 at the top and 100

00:48:19.700 --> 00:48:20.440
at the bottom.

00:48:20.440 --> 00:48:24.070
So it's going to be 100 - 30y.

00:48:24.070 --> 00:48:28.240
We'll let the level be 1, here.

00:48:28.240 --> 00:48:29.500
Sorry, 30.

00:48:29.500 --> 00:48:33.160
I said 3-- I wrote
3, but I meant 30.

00:48:33.160 --> 00:48:39.120
So this is the
situation that we have.

00:48:39.120 --> 00:48:43.280
Now, the point
about this problem

00:48:43.280 --> 00:48:46.170
is we're going to figure
out the total temperature,

00:48:46.170 --> 00:48:48.920
the total amount of heat
that you need to add in order

00:48:48.920 --> 00:48:50.010
to heat this thing up.

00:48:50.010 --> 00:48:53.240
That's going to be
temperature times volume.

00:48:53.240 --> 00:48:56.350
But some places will
count more than others.

00:48:56.350 --> 00:48:59.720
These will be hotter, but
there's less water down here.

00:48:59.720 --> 00:49:02.290
This is wider up here, so
there's more water up here.

00:49:02.290 --> 00:49:05.610
So there are various things that
are varying in this problem.

00:49:05.610 --> 00:49:09.450
Now, the only way to
set up so that it works

00:49:09.450 --> 00:49:13.870
is to chop things
up horizontally

00:49:13.870 --> 00:49:15.780
instead of vertically.

00:49:15.780 --> 00:49:18.800
Because it's on the
horizontal levels

00:49:18.800 --> 00:49:21.090
that the temperature
is constant.

00:49:21.090 --> 00:49:23.010
So we'll have an
easy calculation

00:49:23.010 --> 00:49:28.320
for how much it takes to heat
up a layer, a horizontal layer.

00:49:28.320 --> 00:49:32.850
When we rotate this
guy around the y axis,

00:49:32.850 --> 00:49:36.660
that is which kind of shape.

00:49:36.660 --> 00:49:37.332
It's a disk.

00:49:37.332 --> 00:49:39.540
So actually this one's going
to be an easier problem.

00:49:39.540 --> 00:49:41.820
It's going to be a disk
problem, not a shell problem.

00:49:41.820 --> 00:49:44.111
But we're going to have to
work things out with respect

00:49:44.111 --> 00:49:47.617
to the dy variation.

00:49:47.617 --> 00:49:49.950
In other words, the integral
will be with respect to dy.

00:49:49.950 --> 00:49:51.158
So we will do that next time.

00:49:51.158 --> 00:49:56.410
We'll figure out how much heat
it takes to boil the kettle.