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

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setting up integrals.
 

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And what we'll deal with
is volumes by slices.

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By slicing.
 

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And it's lucky that
this is after lunch.

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Maybe it's after breakfast for
some of you, because there's

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the typical way of introducing
this subject is with

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a food analogy.
 

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There's a lot of ways
of slicing up food.

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And we'll give a few more
examples than just this one.

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But, suppose you have,
well, suppose you have

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a loaf of bread here.
 

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So here's our loaf of bread,
and I hope that looks a little

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bit like a loaf of bread.
 

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It's supposed to be sitting
on the kitchen counter

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ready to be eaten.
 

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And in order to figure out how
much bread there is there, one

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way of doing it is to
cut it into slices.

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Now, you probably know
that bread is often

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sliced like this.
 

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There are even
machines to do it.

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And with this setup here,
I'll draw the slice with

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a little bit of a more
colorful decoration.

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So here's our red
slice of bread.

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It's coming around like this.
 

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And it comes back down behind.
 

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So here's our bread slice.
 

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And what I'd like to
figure out is its volume.

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So first of all, there's the
thickness of the bread.

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Which is this dimension,
the thickness is this

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dimension dx here.
 

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And the only other dimension
that I'm going to give, because

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this is a very qualitative
analysis for now, is what

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I'll call the area.
 

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And that's the area on
the face of the slice.

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And so the area of one slice,
which I'll denote by delta V,

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that's a chunk of volume, is
approximately the area

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times the change in x.
 

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And in the limit, that's going
to be something like this.

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And maybe the areas
of the slices vary.

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There might be a little
hole in the middle of

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the bread somewhere.
 

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Maybe it gets a little
small on one side.

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So it might change
as x changes.

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And the whole volume
you get by adding up.

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So if you like,
this is one slice.

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And this is the sum.
 

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And you should think of it in a
sort of intuitive way as being

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analogous to the Riemann sum,
where you would take each

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slice individually.
 

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And that would look like this.
 

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Alright, so that's just a
superficial and intuitive

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way of looking at it.
 

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Now, we're only going
to talk about one kind

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of systematic slice.
 

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It's already on your problem
set, you had an example of

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a slice of some region.
 

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But we're only going to talk
systematically about something

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called solids of revolution.
 

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The idea here is this.
 

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Suppose you have some
shape, some graph, which

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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.

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In this case, I'm going to
revolve it around the x axis.

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If you do that, then
the shape that you get

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is maybe like this.
 

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If I can draw it a little bit.
 

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It's maybe a football.
 

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So that's the shape that you
get if you take this piece of

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disk and you revolve it around.
 

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If you had made this copy
underneath, it still would

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have been the same region.
 

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So we only pay attention to
what's above the axis here.

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So that's the basic idea.
 

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Now, I'm going to apply the
method of slices to figure out

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the volume of such regions and
give you a general formula.

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And then apply it in
a specific case.

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I want to take one little slice
of this football, maybe a

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football wouldn't work too
well, maybe we should go

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back to a loaf of bread.
 

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Anyway, the key point is
that you never really have

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to draw a 3-D picture.
 

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And 3-D pictures are awful.
 

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They're very hard to deal with.
 

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And it's hard to
visualize with them.

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And one of the reasons why
we're dealing with solids

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of revolution is that
we don't have as many

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visualization problems.
 

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So we're only going
to deal with this.

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And then you have to imagine
from the two-dimensional

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cross-section what the
three-dimensional

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picture looks like.
 

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So we'll do a little bit
of an exercise with the

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three-dimensional picture.
 

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But ultimately, you should be
used to, getting used to,

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drawing 2-D diagrams always.
 

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To depict the
three-dimensional situation.

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Since it's much harder to draw.
 

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The first step is to consider
what this slice is over here.

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And again it's going
to have width dx.

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And we're going to consider
what it looks like over

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on the 3-D picture.
 

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So it starts out being
more or less like this.

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But then we're going
to sweep it around.

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We're revolving around the
x axis, so it's spinning

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around this way.
 

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And if you take this and think
of it as being on a hinge,

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which is down on the x axis,
it's going to swing down and

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swoop around and come back.
 

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Swing around.
 

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And that traces out
something over here.

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Which I'm going to
draw this way.

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It traces out a disk.
 

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So it's hard for me
to draw, and I'm not

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going to try too hard.
 

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Maybe drew it more like a
wheel, looking like a wheel.

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But anyway, it's this
little flat disk here.

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And so the method that I'm
describing for figuring out

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the volume is called
the method of disks.

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This is going to be
our first method.

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Now I'm going to apply the
reasoning that I have up on

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the previous blackboard here.
 

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Namely, I need to get the
volume of this chunk.

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And the way I'm going to get
the volume of this chunk is by

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figuring out its thickness
and its area, its

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cross-sectional area.
 

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And that's not too
difficult to do.

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If this height, so this height
is what we usually call y.

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And y is usually a function
of x, it's varying.

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And this particular
distance is y.

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00:08:31 --> 00:08:35
Then the area of the face
is easy to calculate.

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Because it's a circle,
or a disk if you like,

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with this radius.
 

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So its area is pi y^2.
 

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So that's, if you like,
one of the dimensions.

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And then the thickness is dx.
 

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So the incremental
volume is this.

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00:09:03 --> 00:09:09
So this is the method of disks.
 

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And this is the integrand.
 

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Now, there's one peculiar
thing about this formula.

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And there are more peculiar
things about this formula.

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But there's one peculiar
thing that you should

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notice immediately.
 

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Which is that I'm integrating
with respect to x.

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And I haven't yet
told you what y is.

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Well, that will depend on what
function y = f(x) I use.

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So we have to plug
that in eventually.

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00:09:34 --> 00:09:36
If we're actually going to
calculate something, we're

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going to have to
figure that out.

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00:09:37 --> 00:09:39
There's another very important
point which is that in order to

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get a definite integral,
something I haven't mentioned,

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we're going to have to figure
out where we're starting and

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we're ending the picture.
 

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Which is something we dealt
with last time in 2-D pictures.

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00:09:52 --> 00:09:55
So let's deal with an example.
 

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And we'll switch over from a
football to a soccer ball.

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00:10:05 --> 00:10:10
I'm going to take a circle, and
we'll say it has radius a.

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00:10:10 --> 00:10:13
So this is 0 and this is a.
 

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And I'm putting it in this
particular spot for a reason.

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You can do this in lots of
different ways, but I'm picking

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this one to make a certain
exercise on your homework

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easier for you.
 

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00:10:24 --> 00:10:31
Because I'm doing half of
it for you right now.

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Appreciate it, yeah.
 

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00:10:32 --> 00:10:36
I'm sure especially
today it's appreciated.

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So again, the formula has
to do with keeping track

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of these slices here.
 

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And we're sweeping
things around.

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So the full region that we're
talking about is the volume

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of the ball of radius a.
 

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00:10:57 --> 00:10:59
That's what our goal is, to
figure out what the volume

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00:10:59 --> 00:11:04
of the ball of radius a is.
 

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00:11:04 --> 00:11:09
Alright, again as I say,
the thing sweeps around.

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00:11:09 --> 00:11:15
Coming out of the blackboard,
spinning around on this x axis.

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00:11:15 --> 00:11:18
So the setup is the following.
 

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00:11:18 --> 00:11:21
It's always the same.
 

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Here's our formula.
 

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And we need to figure
out what's going on

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with that formula.
 

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And so we need to solve
for y as a function of x.

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And in order to do that, what
we're going to do is just

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write down the equation
for the circle.

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This is the circle.
 

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00:11:45 --> 00:11:49
It's centered at (a, 0),
so that's its formula.

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00:11:49 --> 00:11:51
And now there's one nice thing,
which is that we really didn't

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need it to find the formula
for y, we only needed to find

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00:11:54 --> 00:11:57
the formula for y squared.
 

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00:11:57 --> 00:12:00
So let's just solve for y^2
and we won't have to solve

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a quadratic or anything.
 

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00:12:01 --> 00:12:06
Take a square root,
that's nice.

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00:12:06 --> 00:12:11
This is a^2 - (x^2
- 2ax + a^2).

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00:12:12 --> 00:12:15
The a^2's cancel.
 

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00:12:15 --> 00:12:18
These two terms cancel.
 

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And so the formula
here is 2ax - x^2.

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Alright now, that is what's
known as the integrant.

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Well, except for this
factor of pi here.

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00:12:38 --> 00:12:41
And so the answer for the
volume is going to be the

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00:12:41 --> 00:12:51
integral of pi times this
integrant, (2ax - x^2) dx.

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00:12:51 --> 00:12:54
And that's just the
same thing as this.

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00:12:54 --> 00:12:57
But now there's also the
issue of the limits.

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00:12:57 --> 00:12:59
Which is a completely
separate problem, which

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we also have to solve.
 

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00:13:03 --> 00:13:08
The range of x is, from
this leftmost point to

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00:13:08 --> 00:13:10
the rightmost point.
 

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00:13:10 --> 00:13:14
So x varies, starts at 0, and
it goes all the way up to what,

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00:13:14 --> 00:13:16
what's the top value here.
 

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00:13:16 --> 00:13:18
2a.
 

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00:13:18 --> 00:13:23
And so now I have a completely
specified integral.

218
00:13:23 --> 00:13:26
Again, and this was the theme
last time, the whole goal is to

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00:13:26 --> 00:13:30
get ourselves to a complete
formula for something with

220
00:13:30 --> 00:13:32
an integrant and limits.
 

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00:13:32 --> 00:13:33
And then we'll be
able to calculate.

222
00:13:33 --> 00:13:40
Now, we have clear sailing
to the end of the problem.

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00:13:40 --> 00:13:42
So let's just finish it off.
 

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00:13:42 --> 00:13:45
We have the volume is, if I
take the antiderivative of

225
00:13:45 --> 00:13:53
that, that's pi ax^2, whose
derivative is 2ax, - x^3 / 3.

226
00:13:53 --> 00:13:57
That's the thing whose
derivative is - x ^2.

227
00:13:57 --> 00:14:02
Evaluated at 0 and 2a.
 

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And that is equal to
pi times, let's see.

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00:14:09 --> 00:14:09
2^2 = 4.
 

230
00:14:09 --> 00:14:19
So this is 4 a^3 -
8a^3 / 3, right?

231
00:14:19 --> 00:14:28
And so all told, that is,
let's see, 12/3 - 8/3 pi a^3.

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00:14:28 --> 00:14:35
Which is maybe a familiar
formula, 4/3 pi a^3.

233
00:14:36 --> 00:14:41
So it worked, we got it right.
 

234
00:14:41 --> 00:14:48
Let me just point out a
couple of other things

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00:14:48 --> 00:14:51
about this formula.
 

236
00:14:51 --> 00:14:59
The first one is that from this
point of view, we've actually

237
00:14:59 --> 00:15:04
accomplished more then just
finding the volume of the ball.

238
00:15:04 --> 00:15:08
We've also found the volume
of a bunch of intermediate

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00:15:08 --> 00:15:12
regions, which I can draw
schematically this way.

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00:15:12 --> 00:15:19
If I chop this thing, and this
portion is x here, then the

241
00:15:19 --> 00:15:24
antiderivative here, this
region here, which maybe I'll

242
00:15:24 --> 00:15:28
fill in with this region here.
 

243
00:15:28 --> 00:15:31
Which I'm going to call V (x).
 

244
00:15:31 --> 00:15:35
Is the volume of the
portion of the sphere.

245
00:15:35 --> 00:15:53
Volume of portion of
width x of the ball.

246
00:15:53 --> 00:15:59
And, well, the formula for
it is that it's the volume

247
00:15:59 --> 00:16:05
= pi (ax ^2 - x ^3 / 3).
 

248
00:16:05 --> 00:16:06
That's it.
 

249
00:16:06 --> 00:16:08
So we've got something
which is actually a

250
00:16:08 --> 00:16:09
lot more information.
 

251
00:16:09 --> 00:16:17
For instance, if you plug in x
= a, not surprisingly, and this

252
00:16:17 --> 00:16:20
is a good idea to do because
it checks that we've actually

253
00:16:20 --> 00:16:23
got a correct formula here.
 

254
00:16:23 --> 00:16:28
So if you like you can
call this a double-check.

255
00:16:28 --> 00:16:32
If you check V ( a), this
should be the volume

256
00:16:32 --> 00:16:36
of a half-ball.
 

257
00:16:36 --> 00:16:38
That's halfway.
 

258
00:16:38 --> 00:16:40
If I go over here and I
only go up to a, that's

259
00:16:40 --> 00:16:42
exactly half of the ball.
 

260
00:16:42 --> 00:16:45
That had better be half,
so let's just see.

261
00:16:45 --> 00:16:48
V ( a) in this case is pi.
 

262
00:16:48 --> 00:16:53
And then I have
(a^3 - a ^3 / 3).

263
00:16:53 --> 00:16:59
And that turns out to be
pi times a total of 2/3 a

264
00:16:59 --> 00:17:07
^3, which is indeed half.
 

265
00:17:07 --> 00:17:11
Now, on your problem set, what
you're going to want to look

266
00:17:11 --> 00:17:17
at is this full formula here.
 

267
00:17:17 --> 00:17:21
Of this chunk.
 

268
00:17:21 --> 00:17:27
And what it's going to be good
for is a real life problem.

269
00:17:27 --> 00:17:31
That is, a problem that really
came up over the summer.

270
00:17:31 --> 00:17:37
And this fall at a couple of
universities near here, where

271
00:17:37 --> 00:17:40
people were trying to
figure out a phenomenon

272
00:17:40 --> 00:17:42
which is well-known.
 

273
00:17:42 --> 00:17:56
Namely, if you have a bunch
of particles in a fluid,

274
00:17:56 --> 00:18:03
and maybe the size of
these things is 1 micron.

275
00:18:03 --> 00:18:06
That is, the radius
is 1 micron.

276
00:18:06 --> 00:18:09
And then you have a bunch
of other little particles,

277
00:18:09 --> 00:18:11
which are a lot smaller.
 

278
00:18:11 --> 00:18:15
Maybe 10 nanometers.
 

279
00:18:15 --> 00:18:23
Then what happens is that the
particles, the big particles,

280
00:18:23 --> 00:18:25
like to hug each other.
 

281
00:18:25 --> 00:18:28
They like to clump together,
they're very nice.

282
00:18:28 --> 00:18:29
Friendly characters.
 

283
00:18:29 --> 00:18:34
So what's the
explanation for this?

284
00:18:34 --> 00:18:39
The explanation is that
actually they are not quite as

285
00:18:39 --> 00:18:40
friendly as they might seem.
 

286
00:18:40 --> 00:18:43
What's really happening is
that the little guys are

287
00:18:43 --> 00:18:45
shoving them around.
 

288
00:18:45 --> 00:18:47
And pushing them together.
 

289
00:18:47 --> 00:18:49
And they have sharp elbows,
the little ones and

290
00:18:49 --> 00:18:50
they're pushing them.
 

291
00:18:50 --> 00:18:53
Don't like them to be
around and they're

292
00:18:53 --> 00:18:55
pushing them together.
 

293
00:18:55 --> 00:18:58
But there's actually
another possibility.

294
00:18:58 --> 00:19:02
Which is that they also
will stick to the sides

295
00:19:02 --> 00:19:06
of the container.
 

296
00:19:06 --> 00:19:09
So there are two things
that actually happen here.

297
00:19:09 --> 00:19:14
And if you want to get a
quantitative handle on how much

298
00:19:14 --> 00:19:18
of this happens, it has to
do with how much space

299
00:19:18 --> 00:19:19
these things take up.
 

300
00:19:19 --> 00:19:23
And so the issue is some kind
of overlap between a band

301
00:19:23 --> 00:19:27
around one sphere and a band
around the other sphere.

302
00:19:27 --> 00:19:32
And this overlap region is
what you have to calculate.

303
00:19:32 --> 00:19:36
You have to calculate
what's in here.

304
00:19:36 --> 00:19:41
And you can do that using
this formula here.

305
00:19:41 --> 00:19:43
It's not even difficult.
 

306
00:19:43 --> 00:19:46
So this is, if you cut it
in half, turns out to

307
00:19:46 --> 00:19:48
be two of these guys.
 

308
00:19:48 --> 00:19:51
And then you're on your way to
figuring out this problem.

309
00:19:51 --> 00:19:53
And the question is,
which do they prefer.

310
00:19:53 --> 00:19:55
Do they prefer to touch each
other, or do they prefer

311
00:19:55 --> 00:19:57
to touch the wall.
 

312
00:19:57 --> 00:19:59
Do they all cluster
to the wall.

313
00:19:59 --> 00:20:02
So you can actually see this
in solutions, what they do.

314
00:20:02 --> 00:20:05
And the question is, to what
extent do they prefer one

315
00:20:05 --> 00:20:07
configuration to the other.
 

316
00:20:07 --> 00:20:11
So that's a real live problem,
which really comes up.

317
00:20:11 --> 00:20:13
Came up just this year.
 

318
00:20:13 --> 00:20:15
And frequently comes up.
 

319
00:20:15 --> 00:20:23
Which is solved by our
first calculation.

320
00:20:23 --> 00:20:27
So now I have, I
called this Method 1.

321
00:20:27 --> 00:20:30
For solids of revolution, which
is called the method of disks.

322
00:20:30 --> 00:20:34
And now I need to tell
you about the other

323
00:20:34 --> 00:20:38
standard method.
 

324
00:20:38 --> 00:21:07
Which is called the
method of shells.

325
00:21:07 --> 00:21:12
So this is our second method.
 

326
00:21:12 --> 00:21:17
I'm going to illustrate
this one with a holiday

327
00:21:17 --> 00:21:19
themed example here.
 

328
00:21:19 --> 00:21:23
This is supposed to be
a witches' cauldron.

329
00:21:23 --> 00:21:28
Whoops, witch, witches, well.
 

330
00:21:28 --> 00:21:33
Maybe more than one witch will
have this cauldron here.

331
00:21:33 --> 00:21:35
So here's this shape.
 

332
00:21:35 --> 00:21:44
And we're going to figure out
how much liquid is in here.

333
00:21:44 --> 00:21:47
I'm going to plot this.
 

334
00:21:47 --> 00:21:54
Maybe I'll put this down
just a bit lower here.

335
00:21:54 --> 00:21:58
And I'm going to
make it a parabola.

336
00:21:58 --> 00:21:59
This is y = x^2.
 

337
00:22:01 --> 00:22:08
And I'm going to make the
top height be y = a.

338
00:22:08 --> 00:22:10
So here's my situation.
 

339
00:22:10 --> 00:22:14
And I want to figure out how
much liquid is in here.

340
00:22:14 --> 00:22:18
Now, the reason why I presented
the problem in this form,

341
00:22:18 --> 00:22:20
of course, is really to get
you used to these things.

342
00:22:20 --> 00:22:23
And the first new thing that I
want you to get used to is the

343
00:22:23 --> 00:22:27
idea that now we can also
revolve around the y axis,

344
00:22:27 --> 00:22:30
not just the x axis.
 

345
00:22:30 --> 00:22:40
So this one is going to be
revolved around the y axis.

346
00:22:40 --> 00:22:41
And that's what happens.
 

347
00:22:41 --> 00:22:44
If you spin the parabola
around, you get this kind

348
00:22:44 --> 00:22:54
of shape, this kind
of solid shape here.

349
00:22:54 --> 00:23:00
Now, I'm going to use the
same kind of slicing

350
00:23:00 --> 00:23:01
that I did before.
 

351
00:23:01 --> 00:23:04
But it's going to look
totally different.

352
00:23:04 --> 00:23:09
Namely, I'll draw
it in red again.

353
00:23:09 --> 00:23:12
I'm going to take a
little slice over here.

354
00:23:12 --> 00:23:17
And now I want to imagine what
happens if it gets revolved

355
00:23:17 --> 00:23:23
around the y axis.
 

356
00:23:23 --> 00:23:24
This time it's not a disk.
 

357
00:23:24 --> 00:23:26
Actually, if I revolve this
around the other way, it

358
00:23:26 --> 00:23:28
would have had a hole in it.
 

359
00:23:28 --> 00:23:30
Which is also possible to do.
 

360
00:23:30 --> 00:23:34
That's practically the same
as the method of disks.

361
00:23:34 --> 00:23:38
You'll maybe discuss
that in recitation.

362
00:23:38 --> 00:23:40
Anyway, we're going to
revolve it around this way.

363
00:23:40 --> 00:23:44
So again, I need to sweep
it around, swing it

364
00:23:44 --> 00:23:47
around like this.
 

365
00:23:47 --> 00:23:54
And I'll draw the shape.
 

366
00:23:54 --> 00:23:56
It's going to sweep around in a
circle and maybe it'll have a

367
00:23:56 --> 00:24:00
little bit of thickness to it.
 

368
00:24:00 --> 00:24:06
And this is the thing that
people call a shell.

369
00:24:06 --> 00:24:12
This is the so-called
shell of the method.

370
00:24:12 --> 00:24:15
I would maybe call it a
cylinder, and another way of

371
00:24:15 --> 00:24:18
thinking about it is that you
can maybe wrap up

372
00:24:18 --> 00:24:22
a piece of paper.
 

373
00:24:22 --> 00:24:27
Like this.
 

374
00:24:27 --> 00:24:29
There it is, there's a
cylinder you can see.

375
00:24:29 --> 00:24:31
Very thin, right?
 

376
00:24:31 --> 00:24:34
Very thin.
 

377
00:24:34 --> 00:24:38
Now, the reason why I used the
piece of paper as an example of

378
00:24:38 --> 00:24:41
this is that I'm going to
have to figure out the

379
00:24:41 --> 00:24:43
volume of this thing.
 

380
00:24:43 --> 00:24:56
Its thickness, as
usual, is equal to dx.

381
00:24:56 --> 00:25:01
Its height is what?
 

382
00:25:01 --> 00:25:05
Well, actually I have to use
the diagram to see that.

383
00:25:05 --> 00:25:15
The top value is a, and the
bottom value is what we call y.

384
00:25:15 --> 00:25:22
So the height is equal
to, I'm sorry, a - y.

385
00:25:22 --> 00:25:25
Now, again this is an
incredibly risky thing here.

386
00:25:25 --> 00:25:27
And I've done this before
I pointed this out on the

387
00:25:27 --> 00:25:29
very first day of lecture.
 

388
00:25:29 --> 00:25:34
The letter y represents a
lot of different things.

389
00:25:34 --> 00:25:39
And in disguise, when I call
this y, I mean y = x^2.

390
00:25:39 --> 00:25:41
In other words, the
interesting curve.

391
00:25:41 --> 00:25:45
I don't mean y = a, which
is the other part.

392
00:25:45 --> 00:25:53
In general, you might think
of it as being equal

393
00:25:53 --> 00:25:57
to y top - y bottom.
 

394
00:25:57 --> 00:26:00
So there are, of course,
two y's involved.

395
00:26:00 --> 00:26:01
In disguise.
 

396
00:26:01 --> 00:26:05
But we have a shorthand, and
the sort of uninteresting one

397
00:26:05 --> 00:26:07
we call by its, we just
evaluate immediately, and

398
00:26:07 --> 00:26:14
the interesting one we
leave as the symbol y.

399
00:26:14 --> 00:26:18
Now, the last bit that I have
to do, I claim, in order to

400
00:26:18 --> 00:26:22
figure this out, is
the circumference.

401
00:26:22 --> 00:26:25
And the reason for that is
that if I think of this

402
00:26:25 --> 00:26:30
thing as like this tube,
or piece of paper here.

403
00:26:30 --> 00:26:33
In order to figure out how much
stuff there is here, all I

404
00:26:33 --> 00:26:39
have to do is unfold it.
 

405
00:26:39 --> 00:26:43
Its size, it's the whole
quantity of paper here is the

406
00:26:43 --> 00:26:44
same whether it's rolled up
like this or whether

407
00:26:44 --> 00:26:46
it's stretched out.
 

408
00:26:46 --> 00:26:56
So if I unwrap it,
it looks like what?

409
00:26:56 --> 00:26:58
Well, it looks like
kind of a slab?

410
00:26:58 --> 00:27:06
Right, it looks like
just a slab like this.

411
00:27:06 --> 00:27:08
And again, the thickness is dx.
 

412
00:27:08 --> 00:27:10
The height is a - y.
 

413
00:27:10 --> 00:27:15
And now we can see that
the length here is

414
00:27:15 --> 00:27:16
all the way around.
 

415
00:27:16 --> 00:27:19
It's the circumference.
 

416
00:27:19 --> 00:27:24
So this is going to be the
circumference when I unwrap it.

417
00:27:24 --> 00:27:26
And in order to figure out
the circumference, I need

418
00:27:26 --> 00:27:29
to figure out the radius.
 

419
00:27:29 --> 00:27:34
So the radius is, on this
diagram, is right over there.

420
00:27:34 --> 00:27:38
This is the radius.
 

421
00:27:38 --> 00:27:42
And that distance is x.
 

422
00:27:42 --> 00:27:45
So this length here is x.
 

423
00:27:45 --> 00:27:56
And so this circumference
is 2 pi x.

424
00:27:56 --> 00:28:00
And this height is still
ay. a - y, sorry.

425
00:28:00 --> 00:28:13
And the thickness is still dx.
 

426
00:28:13 --> 00:28:16
So in total, we're just
going to multiply these

427
00:28:16 --> 00:28:27
numbers together to
get the total volume.

428
00:28:27 --> 00:28:34
We have, in other words, dV =
the product of the (2 pi x)

429
00:28:34 --> 00:28:37
dimension, the (8 - y)
dimension, and the

430
00:28:37 --> 00:28:42
dx dimension.
 

431
00:28:42 --> 00:28:45
Incidentally, dimensional
analysis is very useful and

432
00:28:45 --> 00:28:46
important in these problems.
 

433
00:28:46 --> 00:28:49
You can see that there
are three lengths being

434
00:28:49 --> 00:28:50
multiplied together.
 

435
00:28:50 --> 00:28:52
So we'll get a volume in end.
 

436
00:28:52 --> 00:28:52
Something cubic.
 

437
00:28:52 --> 00:28:55
And we will be coming back to
that, because it's a quite

438
00:28:55 --> 00:28:57
subtle issue sometimes.
 

439
00:28:57 --> 00:29:01
So here's the formula, and
let's simplify it a little bit.

440
00:29:01 --> 00:29:08
We have 2 pi x times, remember,
first I have to substitute,

441
00:29:08 --> 00:29:11
otherwise I'm never going
to be able to integrate.

442
00:29:11 --> 00:29:22
And then I rewrite that as 2 pi
(ax - x^2, whoops, x^3) dx.

443
00:29:22 --> 00:29:25
Better not get that wrong.
 

444
00:29:25 --> 00:29:29
And now the last little bit
here, that I had better be

445
00:29:29 --> 00:29:32
careful about in order to
figure out what the total

446
00:29:32 --> 00:29:37
volume is, is the limits.
 

447
00:29:37 --> 00:29:41
So the volume is going to be
the integral of this quantity

448
00:29:41 --> 00:29:48
2 pi (ax - x ^3) dx.
 

449
00:29:48 --> 00:29:58
And now I have to pay attention
to what the limits are.

450
00:29:58 --> 00:30:05
Now, here you have to be
careful. x is possibly the,

451
00:30:05 --> 00:30:08
you have to always go
back to the 2-D diagram.

452
00:30:08 --> 00:30:10
I went to it immediately,
but that's the whole point.

453
00:30:10 --> 00:30:14
Is that everything gets read
off from this diagram here.

454
00:30:14 --> 00:30:18
When you take this guy and you
sweep it around, you take care

455
00:30:18 --> 00:30:20
of everything that's
to the left.

456
00:30:20 --> 00:30:23
So we only have to count
what's to the right.

457
00:30:23 --> 00:30:24
We don't have to count
anything over here.

458
00:30:24 --> 00:30:27
Because it's taken care
of when we sweep around.

459
00:30:27 --> 00:30:31
So the starting place
is going to be x = 0.

460
00:30:31 --> 00:30:34
That's where we start.
 

461
00:30:34 --> 00:30:41
And where we end is the
farthest, rightmost spot for x.

462
00:30:41 --> 00:30:43
Which is down here.
 

463
00:30:43 --> 00:30:45
You've got to watch out
about where that is.

464
00:30:45 --> 00:30:48
In the y variable,
it's up at y = a.

465
00:30:48 --> 00:30:53
But in the x variable, we
can see that it's what?

466
00:30:53 --> 00:30:56
It's the square root of a.
 

467
00:30:56 --> 00:31:00
So these limits, this is where
you're going to focus all your

468
00:31:00 --> 00:31:02
attention on the integrand
and getting it just right.

469
00:31:02 --> 00:31:05
And then you're going to lose
your steam and not pay

470
00:31:05 --> 00:31:06
attention to the limits.
 

471
00:31:06 --> 00:31:10
They're equally important.
 

472
00:31:10 --> 00:31:11
You've no hope of getting
the right answer without

473
00:31:11 --> 00:31:14
getting the limits right.
 

474
00:31:14 --> 00:31:17
So this is the integral from
0 to square root of a.

475
00:31:17 --> 00:31:25
Again, that's just because
y = a and y = x^2 implies

476
00:31:25 --> 00:31:27
x = square root a.
 

477
00:31:27 --> 00:31:30
That's that upper limit there.
 

478
00:31:30 --> 00:31:34
And now, we're ready
to carry out this, to

479
00:31:34 --> 00:31:35
evaluate this integral.
 

480
00:31:35 --> 00:31:39
So we get 2 - sorry, we get
2 pi ax, that's pi ax^2.

481
00:31:39 --> 00:31:42
 


482
00:31:42 --> 00:31:47
Maybe I'll leave
the 2's in there.

483
00:31:47 --> 00:31:51
2 pi (ax^2 is the
antiderivative of this ax^2 /

484
00:31:51 --> 00:31:56
2, and then here x ^4 / 4),
evaluated at 0 and

485
00:31:56 --> 00:31:59
square root of a.
 

486
00:31:59 --> 00:32:05
And finally, let's
see, what is that.

487
00:32:05 --> 00:32:14
That's 2 pi ( a ^2
/ 2 - a ^2 / 4).

488
00:32:14 --> 00:32:19
Which is a total of 1/4,
right, 2 pi ( a ^2 / 4).

489
00:32:19 --> 00:32:33
Which is pi/ 2 a^2.
 

490
00:32:33 --> 00:32:33
Yes, question.
 

491
00:32:33 --> 00:32:40
STUDENT: [INAUDIBLE]
 

492
00:32:40 --> 00:32:41
PROFESSOR: Right.
 

493
00:32:41 --> 00:32:45
So the question is, why did I
integrate only from the middle

494
00:32:45 --> 00:32:48
to this end, instead of all the
way from over here in minus

495
00:32:48 --> 00:32:52
square root of a, all the
way to the plus end.

496
00:32:52 --> 00:32:55
And the reason is that you have
to look at what's happening

497
00:32:55 --> 00:32:57
with the rotation.
 

498
00:32:57 --> 00:33:01
This red guy, when I swept it
around, I counted the stuff

499
00:33:01 --> 00:33:04
to the right and the left.
 

500
00:33:04 --> 00:33:09
So in other words, if I just
rotate the right half of this,

501
00:33:09 --> 00:33:12
I'm covering the left half.
 

502
00:33:12 --> 00:33:16
So if I counted the stuff from
minus square root of 8, I

503
00:33:16 --> 00:33:17
would be doing it twice.
 

504
00:33:17 --> 00:33:19
I would be doubling
what I need.

505
00:33:19 --> 00:33:21
So it's too much.
 

506
00:33:21 --> 00:33:24
Another way of saying it is if
I wanted to take the whole

507
00:33:24 --> 00:33:29
region, if I rotate it around
only 180 degrees, only pi, that

508
00:33:29 --> 00:33:30
would fill up the whole region.
 

509
00:33:30 --> 00:33:32
If I did both halves.
 

510
00:33:32 --> 00:33:34
And then instead of a
circumference, instead of a

511
00:33:34 --> 00:33:36
2 pi x, I could use a pi x.
 

512
00:33:36 --> 00:33:39
But then I would have
double what I had.

513
00:33:39 --> 00:33:41
So there are two ways
of looking at it.

514
00:33:41 --> 00:33:45
The same goes, actually,
for the football case.

515
00:33:45 --> 00:33:47
When I have that football, I
didn't count the bottom part.

516
00:33:47 --> 00:33:51
Because when I swung it around
the x-axis, the top part

517
00:33:51 --> 00:33:57
sufficed and I could
ignore the bottom half.

518
00:33:57 --> 00:33:59
Yeah, another question.
 

519
00:33:59 --> 00:34:04
STUDENT: [INAUDIBLE]
 

520
00:34:04 --> 00:34:05
PROFESSOR: Ooh, good question.
 

521
00:34:05 --> 00:34:08
The question is, when do you
know, how do you know when to

522
00:34:08 --> 00:34:12
take the rectangle to be
vertical or horizontal.

523
00:34:12 --> 00:34:15
So far we've only done
vertical rectangles.

524
00:34:15 --> 00:34:18
And I'm going to do a
horizontal example in a second.

525
00:34:18 --> 00:34:23
And the answer to the question
of when you do it is this.

526
00:34:23 --> 00:34:26
You can always set
it up both ways.

527
00:34:26 --> 00:34:31
One way may be a difficult
calculation and one way may

528
00:34:31 --> 00:34:33
be an easier calculation.
 

529
00:34:33 --> 00:34:36
Yesterday, we did it - or,
sorry, the last time.

530
00:34:36 --> 00:34:37
Yeah, I guess it was yesterday.
 

531
00:34:37 --> 00:34:41
We did it with the horizontal
and the vertical were quite

532
00:34:41 --> 00:34:42
different in character.
 

533
00:34:42 --> 00:34:43
One of them was really a
mess, and one of them

534
00:34:43 --> 00:34:45
was a little easier.
 

535
00:34:45 --> 00:34:47
So very often, one will be
easier than the other.

536
00:34:47 --> 00:34:51
Every once in a while, one of
them is impossible and the

537
00:34:51 --> 00:34:52
other one is possible.
 

538
00:34:52 --> 00:34:55
In other words, the difference
in difficulty can be extreme.

539
00:34:55 --> 00:34:58
So you don't know
that in advance.

540
00:34:58 --> 00:34:59
Yeah, another question.
 

541
00:34:59 --> 00:35:00
STUDENT: [INAUDIBLE]
 

542
00:35:00 --> 00:35:04
PROFESSOR: The question is, did
we just find the volume when

543
00:35:04 --> 00:35:11
you rotate this green
region around.

544
00:35:11 --> 00:35:17
Or, did we find the volume when
we rotate this whole region.

545
00:35:17 --> 00:35:19
In other words, just the
right half or the right

546
00:35:19 --> 00:35:20
and the left half.
 

547
00:35:20 --> 00:35:21
The answer is both.
 

548
00:35:21 --> 00:35:25
The region that you
get is the same.

549
00:35:25 --> 00:35:27
You always get this cauldron,
whether you take this right

550
00:35:27 --> 00:35:29
half when you rotate it
around or you take both

551
00:35:29 --> 00:35:33
and you rotate it around.
 

552
00:35:33 --> 00:35:34
So the answer to both of
those questions is the

553
00:35:34 --> 00:35:35
same and it's this.
 

554
00:35:35 --> 00:35:37
Yes.
 

555
00:35:37 --> 00:35:43
STUDENT: [INAUDIBLE]
 

556
00:35:43 --> 00:35:45
PROFESSOR: So that if you
rotated them both around and

557
00:35:45 --> 00:35:47
you only wanted to cover
things once, you would

558
00:35:47 --> 00:35:49
rotate halfway around.
 

559
00:35:49 --> 00:35:52
Only by 180 degrees.
 

560
00:35:52 --> 00:35:53
That's true.
 

561
00:35:53 --> 00:35:55
But you can rotate around
as many times as you want.

562
00:35:55 --> 00:35:56
You're still covering
the same thing.

563
00:35:56 --> 00:36:03
Over and over and
over and over again.

564
00:36:03 --> 00:36:05
So let's go on.
 

565
00:36:05 --> 00:36:09
I have a very subtle point that
I need to discuss with you in

566
00:36:09 --> 00:36:23
order to go on to the
next application.

567
00:36:23 --> 00:36:27
So here's my, and I do want to
get to the point of horizontal

568
00:36:27 --> 00:36:29
cross-sections as well.
 

569
00:36:29 --> 00:36:32
So let's continue here.
 

570
00:36:32 --> 00:36:38
So the first thing that I want
to point out to you now is, I

571
00:36:38 --> 00:36:50
want you to beware of units.
 

572
00:36:50 --> 00:36:54
There's something a little
fishy in this problem.

573
00:36:54 --> 00:37:01
And it can be summarized in the
character of the answer, which

574
00:37:01 --> 00:37:05
is just a little bit not clear.
 

575
00:37:05 --> 00:37:09
Namely, it looks like
it's a^2, right?

576
00:37:09 --> 00:37:12
And we know that a is in
units and we should've

577
00:37:12 --> 00:37:14
gotten cubic units.
 

578
00:37:14 --> 00:37:18
So there's something a little
bit tricky about this question.

579
00:37:18 --> 00:37:22
And so I want to illustrate
the paradox right now.

580
00:37:22 --> 00:37:26
So, suppose that a
= 100 centimeters.

581
00:37:26 --> 00:37:29
And suppose the units
are centimeters.

582
00:37:29 --> 00:37:35
Then the formula for the
volume is pi / 2 ( 100)^2.

583
00:37:36 --> 00:37:40
and the units we must take
are centimeters cubed.

584
00:37:40 --> 00:37:42
Despite the fact that you
kind of want to square

585
00:37:42 --> 00:37:43
the centimeters.
 

586
00:37:43 --> 00:37:45
But that's not what
this problem says.

587
00:37:45 --> 00:37:50
OK, so this is the
situation that we've got.

588
00:37:50 --> 00:37:55
Now, if you work out what this
is, to figure out what the

589
00:37:55 --> 00:38:01
volume of this cauldron is,
what you find is that it's pi /

590
00:38:01 --> 00:38:10
2 times, well, 10^ 4 is 10
* 1,000 centimeters cubed.

591
00:38:10 --> 00:38:12
And those are otherwise
known as liters.

592
00:38:12 --> 00:38:15
So this is approximately
10 pi / 2 liters, which

593
00:38:15 --> 00:38:20
is about 16 liters.
 

594
00:38:20 --> 00:38:23
And so that's how much was
in the cauldron under

595
00:38:23 --> 00:38:27
this choice of units.
 

596
00:38:27 --> 00:38:30
Now, I'm going to make
another choice of units now.

597
00:38:30 --> 00:38:32
And we're going to
make a comparison.

598
00:38:32 --> 00:38:38
Suppose that the
units are 1 meter.

599
00:38:38 --> 00:38:42
Looks like it should be the
same, but if I calculate the

600
00:38:42 --> 00:38:50
volume, it's going to be
pi / 2, and 1^2 * m^3.

601
00:38:51 --> 00:38:55
That's what the formula
tells us to do.

602
00:38:55 --> 00:39:04
And if you calculate that out,
that's pi / 2 (100 cm)^3.

603
00:39:04 --> 00:39:10
And with this unit notation we
really do want to cube the

604
00:39:10 --> 00:39:12
centimeters and cube the 100.
 

605
00:39:12 --> 00:39:14
So we get pi / 2.
 

606
00:39:14 --> 00:39:17
And here we get 10 ^ 6 cm^3.
 

607
00:39:18 --> 00:39:26
And that comes out to
pi / 2 * 1000 liters.

608
00:39:26 --> 00:39:35
Or, in other words,
about 1600 liters.

609
00:39:35 --> 00:39:40
So I'd like to ask you
first to contemplate this.

610
00:39:40 --> 00:39:42
And this is a paradox.
 

611
00:39:42 --> 00:39:44
And this is a serious paradox.
 

612
00:39:44 --> 00:39:46
If you really want to apply
problems, you actually have

613
00:39:46 --> 00:39:49
to understand what
your answers mean.

614
00:39:49 --> 00:39:53
So what do you think
is going on here?

615
00:39:53 --> 00:39:53
Yeah.
 

616
00:39:53 --> 00:40:07
STUDENT: [INAUDIBLE]
 

617
00:40:07 --> 00:40:07
PROFESSOR: Yes.
 

618
00:40:07 --> 00:40:19
STUDENT: [INAUDIBLE]
 

619
00:40:19 --> 00:40:19
PROFESSOR: Right.
 

620
00:40:19 --> 00:40:22
So the question is, how could
either of these make sense.

621
00:40:22 --> 00:40:26
How am I dealing with the
units in either case.

622
00:40:26 --> 00:40:30
So now I'm going to explain
to you the answer.

623
00:40:30 --> 00:40:32
Because this is really
quite puzzling.

624
00:40:32 --> 00:40:36
But it has a resolution.
 

625
00:40:36 --> 00:40:49
The answer to this question is
that both answers are correct.

626
00:40:49 --> 00:40:54
This is correct reasoning
in both cases.

627
00:40:54 --> 00:41:00
What's the matter is that you
have to interpret the equation

628
00:41:00 --> 00:41:13
y = x ^2 in two different ways.
 

629
00:41:13 --> 00:41:17
Two ways.
 

630
00:41:17 --> 00:41:19
So let me explain
what they are.

631
00:41:19 --> 00:41:31
For instance, you can take
y = x^2 in centimeters.

632
00:41:31 --> 00:41:35
So y = x ^2 in centimeters.
 

633
00:41:35 --> 00:41:44
In which case, the picture
looks like the following.

634
00:41:44 --> 00:41:49
a = 100 centimeters.
 

635
00:41:49 --> 00:41:55
And this distance here, which
is the x value, this is 10.

636
00:41:55 --> 00:41:57
This is 10 centimeters.
 

637
00:41:57 --> 00:41:59
And that's what the
relationship means.

638
00:41:59 --> 00:42:03
So the top of the cauldron,
if you like, this distance

639
00:42:03 --> 00:42:06
here is 20 centimeters.
 

640
00:42:06 --> 00:42:08
This is actually very
badly drawn to scale.

641
00:42:08 --> 00:42:10
It's actually very,
very, deep, this thing.

642
00:42:10 --> 00:42:13
It's a rather skinny, deep one.
 

643
00:42:13 --> 00:42:19
So this is very much not
to scale, this picture.

644
00:42:19 --> 00:42:26
The other picture, the other
picture is interpreting

645
00:42:26 --> 00:42:34
y = x ^2 in meters.
 

646
00:42:34 --> 00:42:37
And that's more like what
I had in mind, actually.

647
00:42:37 --> 00:42:40
I had in mind this
big vat here.

648
00:42:40 --> 00:42:47
And this distance here is
1 meter, and then the

649
00:42:47 --> 00:42:49
square root of 1 is 1.
 

650
00:42:49 --> 00:42:52
So this distance here
is also 1 meter.

651
00:42:52 --> 00:43:00
And the top is 2 meters.
 

652
00:43:00 --> 00:43:02
Now, it's not that crazy.
 

653
00:43:02 --> 00:43:05
And in fact it's easy to check
that, it's pretty reasonable

654
00:43:05 --> 00:43:06
in terms of scale.
 

655
00:43:06 --> 00:43:09
That this thing has
16 liters in it.

656
00:43:09 --> 00:43:17
And this guy has
1600 liters in it.

657
00:43:17 --> 00:43:20
So you actually have to know
what your symbols mean when

658
00:43:20 --> 00:43:23
you're dealing with these
kinds of applied problems.

659
00:43:23 --> 00:43:27
And if you're ever really going
to do some real consequences,

660
00:43:27 --> 00:43:30
you have to know what
the units mean.

661
00:43:30 --> 00:43:32
And the problem with the
equation y = x^2 is

662
00:43:32 --> 00:43:37
that it's the one that
violated scaling rules.

663
00:43:37 --> 00:43:38
Yeah.
 

664
00:43:38 --> 00:43:47
STUDENT: [INAUDIBLE]
 

665
00:43:47 --> 00:43:47
PROFESSOR: Yeah.
 

666
00:43:47 --> 00:43:53
STUDENT: [INAUDIBLE]
 

667
00:43:53 --> 00:43:53
PROFESSOR: No.
 

668
00:43:53 --> 00:43:59
STUDENT: [INAUDIBLE]
 

669
00:43:59 --> 00:44:06
PROFESSOR: OK, so the question
is whether the formula

670
00:44:06 --> 00:44:11
V = pi / 2a ^2.
 

671
00:44:11 --> 00:44:14
This is the correct
answer to the problem.

672
00:44:14 --> 00:44:17
But it is not
consistent in units.

673
00:44:17 --> 00:44:21
If you plug in a equals some
number of centimeters, some

674
00:44:21 --> 00:44:24
number of millimeters, some
number of inches and so on,

675
00:44:24 --> 00:44:26
every single time you'll
get a different answer.

676
00:44:26 --> 00:44:30
And they're all inconsistent.
 

677
00:44:30 --> 00:44:33
In other words, this
formula violates scaling.

678
00:44:33 --> 00:44:36
STUDENT: [INAUDIBLE]
 

679
00:44:36 --> 00:44:44
PROFESSOR: If you study each
step correctly, you will

680
00:44:44 --> 00:44:47
discover that these are the
consistent and correct

681
00:44:47 --> 00:44:49
statements, what I'm
writing on the blackboard.

682
00:44:49 --> 00:44:53
And this makes sense
in a unit-less sense.

683
00:44:53 --> 00:44:56
But then if you actually stick
units on them, one of them,

684
00:44:56 --> 00:44:58
they both are correct.
 

685
00:44:58 --> 00:45:00
And one of them describes this
situation and one of them

686
00:45:00 --> 00:45:03
describes this situation.
 

687
00:45:03 --> 00:45:07
And it's a mistake to think
of this as being 1 meter

688
00:45:07 --> 00:45:09
and cubing the meters.
 

689
00:45:09 --> 00:45:20
That will be an error that
will cause you problems.

690
00:45:20 --> 00:45:23
This 1 is just unit-less,
and then the meters

691
00:45:23 --> 00:45:30
cubed got converted.
 

692
00:45:30 --> 00:45:33
So I encourage you to
study this on your own.

693
00:45:33 --> 00:45:35
So now I'm going to
introduce the next problem.

694
00:45:35 --> 00:45:38
We'll have to solve
it next time.

695
00:45:38 --> 00:45:41
But the reason why I spent all
the time on units is that

696
00:45:41 --> 00:45:44
otherwise it would be
impossible for you to believe

697
00:45:44 --> 00:45:47
me when I do this
next calculation.

698
00:45:47 --> 00:45:48
Because we're trying to
get a real answer out

699
00:45:48 --> 00:45:50
of a real question.
 

700
00:45:50 --> 00:45:53
And I'm going to make
conversions between centimeters

701
00:45:53 --> 00:45:55
and meters back and forth.
 

702
00:45:55 --> 00:45:57
And we have to get it
consistent in order to

703
00:45:57 --> 00:45:58
have the right answer.
 

704
00:45:58 --> 00:46:04
So there was a reason for
illustrating this pitfall.

705
00:46:04 --> 00:46:11
So this second, the next thing
that I'd like to do, is I'd

706
00:46:11 --> 00:46:19
like to boil the water in
the witches' cauldron.

707
00:46:19 --> 00:46:23
This is definitely seasonally
appropriate, since we're

708
00:46:23 --> 00:46:25
approaching Halloween.
 

709
00:46:25 --> 00:46:29
And we'll work it out
fully next time.

710
00:46:29 --> 00:46:35
Now, I'm going to
introduce another feature

711
00:46:35 --> 00:46:36
into the problem.
 

712
00:46:36 --> 00:46:38
And this is the one that I
want you to understand now.

713
00:46:38 --> 00:46:41
We'll set it all up tomorrow,
but right now I need you to

714
00:46:41 --> 00:46:51
understand what the new main
idea that we're going to get.

715
00:46:51 --> 00:46:54
There is the new physical
feature that I'm going to add

716
00:46:54 --> 00:46:57
to this problem is that if when
you're boiling, when the

717
00:46:57 --> 00:47:02
witches are boiling this water,
the temperature of the water is

718
00:47:02 --> 00:47:06
not the same at each
level in the kettle.

719
00:47:06 --> 00:47:08
At the bottom of the kettle,
where you're heating it up,

720
00:47:08 --> 00:47:11
it's at its highest
temperature.

721
00:47:11 --> 00:47:15
So at the bottom it's going
to be, say, 100 degrees.

722
00:47:15 --> 00:47:17
That is, it's going to
be totally boiling.

723
00:47:17 --> 00:47:18
100 degrees Celsius.
 

724
00:47:18 --> 00:47:30
And at the top, it's going
to be, say, 70 degrees.

725
00:47:30 --> 00:47:32
Right, it's very cold outside.
 

726
00:47:32 --> 00:47:37
In fact, it's 0
degrees outside.

727
00:47:37 --> 00:47:40
Which is the temperature
at which all witches

728
00:47:40 --> 00:47:41
operate, I think.
 

729
00:47:41 --> 00:47:46
Anyway, so they're
boiling their stuff.

730
00:47:46 --> 00:47:49
And the question that we're
going to ask is how much

731
00:47:49 --> 00:47:55
heat, how much heat,
do they need to do it.

732
00:47:55 --> 00:48:00
Now, the thing starts out
at 0 degrees Celsius.

733
00:48:00 --> 00:48:01
And we're going to heat it
up to this temperature

734
00:48:01 --> 00:48:03
configuration here.
 

735
00:48:03 --> 00:48:07
But it's rising from 100
down here to 70 up here.

736
00:48:07 --> 00:48:10
So the temperature is
varying in height.

737
00:48:10 --> 00:48:13
And for simplicity I'm going
to make the formula for the

738
00:48:13 --> 00:48:20
temperature be 70 at the
top and 100 at the bottom.

739
00:48:20 --> 00:48:24
So it's going to be 100 - 30y.
 

740
00:48:24 --> 00:48:28
We'll let the level be 1, here.
 

741
00:48:28 --> 00:48:29
Sorry, 30.
 

742
00:48:29 --> 00:48:33
I said 3, I wrote
3, but I meant 30.

743
00:48:33 --> 00:48:39
So this is the situation
that we have.

744
00:48:39 --> 00:48:44
Now, the point about this
problem is we're going to

745
00:48:44 --> 00:48:46
figure out the total
temperature, the total amount

746
00:48:46 --> 00:48:50
of heat that you need to add in
order to heat this thing up.

747
00:48:50 --> 00:48:53
That's going to be
temperature times volume.

748
00:48:53 --> 00:48:56
But some places will
count more than others.

749
00:48:56 --> 00:48:59
These will be hotter, but
there's less water down here.

750
00:48:59 --> 00:49:02
This is wider up here, so
there's more water up here.

751
00:49:02 --> 00:49:04
So there are various
things that are varying

752
00:49:04 --> 00:49:05
in this problem.
 

753
00:49:05 --> 00:49:11
Now, the only way to set up
so that it works is to chop

754
00:49:11 --> 00:49:15
things up horizontally
instead of vertically.

755
00:49:15 --> 00:49:19
Because it's on the
horizontal levels that the

756
00:49:19 --> 00:49:21
temperature is constant.
 

757
00:49:21 --> 00:49:23
So we'll have an easy
calculation for how much it

758
00:49:23 --> 00:49:28
takes to heat up a layer,
a horizontal layer.

759
00:49:28 --> 00:49:33
When we rotate this guy
around the y axis, that

760
00:49:33 --> 00:49:36
is which kind of shape.
 

761
00:49:36 --> 00:49:37
It's a disk.
 

762
00:49:37 --> 00:49:39
So actually this one's going
to be an easier problem.

763
00:49:39 --> 00:49:41
It's going to be a disk
problem, not a shell problem.

764
00:49:41 --> 00:49:43
But we're going to have to
work things out with respect

765
00:49:43 --> 00:49:47
to the dy variation.
 

766
00:49:47 --> 00:49:49
In other words, the integral
will be with respect to dy.

767
00:49:49 --> 00:49:51
So we will do that next time.
 

768
00:49:51 --> 00:49:56
We'll figure out how much heat
it takes to boil the kettle.

769
00:49:56 --> 00:49:57