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All right, so the past few
weeks,
we've been looking at double
integrals and the plane,

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line integrals in the plane,
and will we are going to do now

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from now on basically until the
end of the term,

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00:00:34,000 --> 00:00:36,000
will be very similar stuff,
but in space.

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So, we are going to learn how
to do triple integrals in space,

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flux in space,
work in space,

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divergence, curl,
all that.

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00:00:46,000 --> 00:00:49,000
So,
that means, basically,

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00:00:49,000 --> 00:00:52,000
if you were really on top of
what we've been doing these past

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few weeks,
then it will be just the same

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with one more coordinate.
And, you will see there are

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00:00:58,000 --> 00:01:00,000
some differences.
But, conceptually,

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it's pretty similar.
There are a few tricky things,

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00:01:04,000 --> 00:01:06,000
though.
Now, that also means that if

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00:01:06,000 --> 00:01:10,000
there is stuff that you are not
sure about in the plane,

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00:01:10,000 --> 00:01:14,000
then I encourage you to review
the material that we've done

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over the past few weeks to make
sure that everything in the

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plane is completely clear to you
because it will be much harder

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00:01:22,000 --> 00:01:26,000
to understand stuff in space if
things are still shaky in the

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plane.
OK, so the plan is we're going

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to basically go through the same
stuff, but in space.

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00:01:36,000 --> 00:01:45,000
So, it shouldn't be surprising
that we will start today with

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00:01:45,000 --> 00:01:52,000
triple integrals.
OK, so the way triple integrals

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00:01:52,000 --> 00:01:58,000
work is if I give you a function
of three variables,

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00:01:58,000 --> 00:02:02,000
x, y, z,
and I give you some region in

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00:02:02,000 --> 00:02:07,000
space,
so, some solid,

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00:02:07,000 --> 00:02:15,000
then I can take the integral
over this region over function f

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00:02:15,000 --> 00:02:20,000
dV where dV stands for the
volume element.

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00:02:20,000 --> 00:02:24,000
OK, so what it means is we will
just take every single little

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00:02:24,000 --> 00:02:28,000
piece of our solid,
take the value of f there,

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00:02:28,000 --> 00:02:30,000
multiply by the small volume of
each little piece,

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00:02:30,000 --> 00:02:35,000
and sum all these things
together.

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00:02:35,000 --> 00:02:40,000
And,
so this volume element here,

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00:02:40,000 --> 00:02:44,000
well, for example,
if you are doing the integral

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00:02:44,000 --> 00:02:48,000
in rectangular coordinates,
that will become dx dy dz or

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00:02:48,000 --> 00:02:54,000
any permutation of that because,
of course, we have lots of

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00:02:54,000 --> 00:02:59,000
possible orders of integration
to choose from.

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00:02:59,000 --> 00:03:08,000
So, rather than bore you with
theory and all sorts of

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00:03:08,000 --> 00:03:15,000
complicated things,
let's just do examples.

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00:03:15,000 --> 00:03:18,000
And, you will see, basically,
if you understand how to set up

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00:03:18,000 --> 00:03:21,000
iterated integrals into
variables,

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00:03:21,000 --> 00:03:23,000
that you basically understand
how to do them in three

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variables.
You just have to be a bit more

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00:03:27,000 --> 00:03:30,000
careful.
And, there's one more step.

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00:03:30,000 --> 00:03:35,000
OK, so let's take our first
triple integral to be on the

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region.
So, of course,

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00:03:36,000 --> 00:03:37,000
there's two different things as
always.

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00:03:37,000 --> 00:03:39,000
There is the region of
integration and there's the

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00:03:39,000 --> 00:03:42,000
function we are integrating.
Now, the function we are

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00:03:42,000 --> 00:03:44,000
integrating, well,
it will come in handy when you

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00:03:44,000 --> 00:03:46,000
actually try to evaluate the
integral.

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00:03:46,000 --> 00:03:49,000
But, as you can see,
probably, the new part is

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really hard to set it up.
So, the function won't really

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00:03:52,000 --> 00:03:55,000
matter that much for me.
So, in the examples I'll do

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today, functions will be kind of
silly.

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00:03:58,000 --> 00:04:04,000
So, for example,
let's say that we want to look

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00:04:04,000 --> 00:04:13,000
at the region between two
paraboloids, one given by z = x

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00:04:13,000 --> 00:04:20,000
^2 y ^2.
The other is z = 4 - x ^2 - y

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00:04:20,000 --> 00:04:22,000
^2.
And, so, I haven't given you,

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00:04:22,000 --> 00:04:26,000
yet, the function to integrate.
OK, this is not the function to

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integrate.
This is what describes the

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00:04:28,000 --> 00:04:32,000
region where I will integrate my
function.

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00:04:32,000 --> 00:04:38,000
And, let's say that I just want
to find the volume of this

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00:04:38,000 --> 00:04:43,000
region, which is the triple
integral of just one dV.

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00:04:43,000 --> 00:04:46,000
OK, similarly,
remember, when we try to find

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00:04:46,000 --> 00:04:49,000
the area of the region in the
plane, we are just integrating

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00:04:49,000 --> 00:04:51,000
one dA.
Here we integrate one dV.

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00:04:51,000 --> 00:04:55,000
that will give us the volume.
Now, I know that you can

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00:04:55,000 --> 00:04:59,000
imagine how to actually do this
one as a double integral.

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00:04:59,000 --> 00:05:02,000
But, the goal of the game is to
set up the triple integral.

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00:05:02,000 --> 00:05:05,000
It's not actually to find the
volume.

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00:05:05,000 --> 00:05:12,000
So, what does that look like?
Well, z = x ^2 y ^2,

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00:05:12,000 --> 00:05:16,000
that's one of our favorite
paraboloids.

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00:05:16,000 --> 00:05:22,000
That's something that looks
like a parabola with its bottom

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00:05:22,000 --> 00:05:28,000
at the origin that you spin
about the z axis.

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00:05:28,000 --> 00:05:32,000
And, z equals four minus x
squared minus y squared,

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00:05:32,000 --> 00:05:36,000
well, that's also a paraboloid.
But, this one is pointing down,

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00:05:36,000 --> 00:05:40,000
and when you take x equals y
equals zero, you get z equals

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00:05:40,000 --> 00:05:44,000
four.
So, it starts at four,

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00:05:44,000 --> 00:05:52,000
and it goes down like that.
OK, so the solid that we'd like

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00:05:52,000 --> 00:05:56,000
to consider is what's in between
in here.

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00:05:56,000 --> 00:06:00,000
So, it has a curvy top which is
this downward paraboloid,

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00:06:00,000 --> 00:06:04,000
a curvy bottom which is the
other paraboloid.

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00:06:04,000 --> 00:06:08,000
And, what about the sides?
Well, do you have any idea what

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00:06:08,000 --> 00:06:11,000
we get here?
Yeah, it's going to be a circle

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00:06:11,000 --> 00:06:15,000
because entire picture is
invariant by rotation about the

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z axis.
So, if you look at the picture

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00:06:17,000 --> 00:06:20,000
just, say, in the yz plane,
you get this point and that

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00:06:20,000 --> 00:06:24,000
point.
And, when you rotate everything

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00:06:24,000 --> 00:06:30,000
around the z axis,
you will just get a circle

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00:06:30,000 --> 00:06:33,000
here.
OK, so our goal is to find the

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00:06:33,000 --> 00:06:36,000
volume of this thing,
and there's lots of things I

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00:06:36,000 --> 00:06:38,000
could do to simplify the
calculation,

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00:06:38,000 --> 00:06:41,000
or even not do it as a triple
integral at all.

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00:06:41,000 --> 00:06:46,000
But, I want to actually set it
up as a triple integral just to

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show how we do that.
OK, so the first thing we need

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00:06:50,000 --> 00:06:54,000
to do is choose an order of
integration.

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00:06:54,000 --> 00:06:56,000
And, here, well,
I don't know if you can see it

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00:06:56,000 --> 00:07:00,000
yet, but hopefully soon that
will be intuitive to you.

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00:07:00,000 --> 00:07:04,000
I claim that I would like to
start by integrating first over

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00:07:04,000 --> 00:07:05,000
z.
What's the reason for that?

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00:07:05,000 --> 00:07:09,000
Well, the reason is if I give
you x and y, then you can find

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00:07:09,000 --> 00:07:13,000
quickly, what's the bottom and
top values of z for that choice

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00:07:13,000 --> 00:07:18,000
of x and y?
OK, so if I have x and y given,

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00:07:18,000 --> 00:07:25,000
then I can find above that:
what is the bottom z and the

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00:07:25,000 --> 00:07:33,000
top z corresponding to the
vertical line above that point?

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00:07:33,000 --> 00:07:39,000
The portion of it that's inside
our solid, so somehow,

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00:07:39,000 --> 00:07:45,000
there's a bottom z and a top z.
And, so the top z is actually

115
00:07:45,000 --> 00:07:49,000
on the downward paraboloid.
So, it's four minus x squared

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00:07:49,000 --> 00:07:52,000
minus y squared.
The bottom value of z is x

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00:07:52,000 --> 00:07:58,000
squared plus y squared.
OK, so if I want to start to

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00:07:58,000 --> 00:08:04,000
set this up, I will write the
triple integral.

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00:08:04,000 --> 00:08:09,000
And then, so let's say I'm
going to do it dz first,

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00:08:09,000 --> 00:08:11,000
and then, say,
dy dx.

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00:08:11,000 --> 00:08:16,000
It doesn't really matter.
So then, for a given value of x

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00:08:16,000 --> 00:08:19,000
and y, I claim z goes from the
bottom surface.

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The bottom face is z equals x
squared plus y squared.

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00:08:23,000 --> 00:08:29,000
The top face is four minus x
squared minus y squared.

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00:08:29,000 --> 00:08:35,000
OK, is that OK with everyone?
Yeah?

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00:08:35,000 --> 00:08:43,000
Any questions so far?
Yes?

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00:08:43,000 --> 00:08:45,000
Why did I start with z?
That's a very good question.

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00:08:45,000 --> 00:08:49,000
So, I can choose whatever order
I want, but let's say I did x

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00:08:49,000 --> 00:08:50,000
first .
Then, to find the inner

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00:08:50,000 --> 00:08:53,000
integral bounds,
I would need to say, OK,

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00:08:53,000 --> 00:08:56,000
I've chosen values of,
see, in the inner integral,

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00:08:56,000 --> 00:08:59,000
you've fixed the two other
variables,

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00:08:59,000 --> 00:09:01,000
and you're just going to vary
that one.

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00:09:01,000 --> 00:09:02,000
And, you need to find bounds
for it.

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00:09:02,000 --> 00:09:05,000
So, if I integrate over x
first, I have to solve,

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00:09:05,000 --> 00:09:10,000
answer the following question.
Say I'm given values of y and z.

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00:09:10,000 --> 00:09:14,000
What are the bounds for x?
So, that would mean I'm slicing

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00:09:14,000 --> 00:09:18,000
my solid by lines that are
parallel to the x axis.

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00:09:18,000 --> 00:09:21,000
And, see, it's kind of hard to
find, what are the values of x

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00:09:21,000 --> 00:09:24,000
at the front and at the back?
I mean, it's possible,

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00:09:24,000 --> 00:09:27,000
but it's easier to actually
first look for z at the top and

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00:09:27,000 --> 00:09:33,000
bottom.
Yes?

143
00:09:33,000 --> 00:09:36,000
dy dx, or dx dy?
No, it's completely at random.

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00:09:36,000 --> 00:09:39,000
I mean, you can see x and y
play symmetric roles.

145
00:09:39,000 --> 00:09:43,000
So, if you look at it,
it's reasonably clear that z

146
00:09:43,000 --> 00:09:49,000
should be the easiest one to set
up first for what comes next.

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00:09:49,000 --> 00:09:54,000
xy or yx, it's the same.
Yes?

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00:09:54,000 --> 00:09:56,000
Yes, it will be easier to use
cylindrical coordinates.

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00:09:56,000 --> 00:10:03,000
I'll get to that just as soon
as I'm done with this one.

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00:10:03,000 --> 00:10:07,000
OK, so let's continue a bit
with that.

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00:10:07,000 --> 00:10:11,000
And, as you mentioned,
actually we don't actually want

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00:10:11,000 --> 00:10:14,000
to do it with xy in the end.
In a few minutes,

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00:10:14,000 --> 00:10:16,000
we will actually switch to
cylindrical coordinates.

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00:10:16,000 --> 00:10:18,000
But, for now,
we don't even know what they

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00:10:18,000 --> 00:10:20,000
are.
OK, so I've done the inner

156
00:10:20,000 --> 00:10:25,000
integral by looking at,
you know, if I slice by

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00:10:25,000 --> 00:10:28,000
vertical lines,
what is the top?

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00:10:28,000 --> 00:10:31,000
What is the bottom for a given
value of x and y?

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00:10:31,000 --> 00:10:36,000
So, the bounds in the inner
integral depend on both the

160
00:10:36,000 --> 00:10:41,000
middle and outer variables.
Next, I need to figure out what

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00:10:41,000 --> 00:10:44,000
values of x and y I will be
interested in.

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00:10:44,000 --> 00:10:47,000
And, the answer for that is,
well, the values of x and y

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00:10:47,000 --> 00:10:51,000
that I want to look at are all
those that are in the shade of

164
00:10:51,000 --> 00:10:53,000
my region.
So, in fact,

165
00:10:53,000 --> 00:10:57,000
to set up the middle and outer
bounds, what I want to do is

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00:10:57,000 --> 00:11:04,000
project my solid.
So, my solid looks like this

167
00:11:04,000 --> 00:11:09,000
kind of thing.
And, I don't really know how to

168
00:11:09,000 --> 00:11:13,000
call it.
But, what's interesting now is

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00:11:13,000 --> 00:11:18,000
I want to look at the shadow
that it casts in the xy plane.

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00:11:18,000 --> 00:11:22,000
OK, and, of course,
that shadow will just be the

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00:11:22,000 --> 00:11:27,000
disk that's directly below this
disk here that's separating the

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00:11:27,000 --> 00:11:34,000
two halves of the solid.
And so, now I will want to

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00:11:34,000 --> 00:11:41,000
integrate over,
I want to look at all the xy's,

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00:11:41,000 --> 00:11:46,000
x and y, in the shadow.
So, now I'm left with,

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00:11:46,000 --> 00:11:48,000
actually, something we've
already done,

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00:11:48,000 --> 00:11:52,000
namely setting up a double
integral over x and y.

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00:11:52,000 --> 00:11:55,000
So, if it helps,
here, we don't strictly need

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00:11:55,000 --> 00:11:59,000
it, but if it helps,
it could be useful to actually

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00:11:59,000 --> 00:12:03,000
draw a picture of this shadow in
the xy plane.

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00:12:03,000 --> 00:12:14,000
So, here it would just look,
again, like a disk,

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00:12:14,000 --> 00:12:18,000
and set it up.
Now, the question is,

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00:12:18,000 --> 00:12:22,000
how do we find the size of this
disk, the size of the shadow?

183
00:12:22,000 --> 00:12:28,000
Well, basically we have to
figure out where our two

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00:12:28,000 --> 00:12:37,000
paraboloids intersect.
There's nothing else.

185
00:12:37,000 --> 00:12:49,000
OK, so, one way how to find the
shadow in the xy plane -- --

186
00:12:49,000 --> 00:12:53,000
well,
here we actually know the

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00:12:53,000 --> 00:12:56,000
answer a priori,
but even if we didn't,

188
00:12:56,000 --> 00:12:59,000
we could just say,
well, our region lives wherever

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00:12:59,000 --> 00:13:03,000
the bottom surface is below the
top surface,

190
00:13:03,000 --> 00:13:11,000
OK, so we want to look at
things wherever bottom value of

191
00:13:11,000 --> 00:13:15,000
z is less than the top value of
z,

192
00:13:15,000 --> 00:13:18,000
I mean, less or less than or
equal, that's the same thing.

193
00:13:18,000 --> 00:13:24,000
So, if the bottom value of z is
x squared plus y squared should

194
00:13:24,000 --> 00:13:28,000
be less than four minus x
squared minus y squared,

195
00:13:28,000 --> 00:13:33,000
and if you solve for that,
then you will get,

196
00:13:33,000 --> 00:13:34,000
well, so let's move these guys
over here.

197
00:13:34,000 --> 00:13:37,000
You'll get two x squared plus
two y squared less than four.

198
00:13:37,000 --> 00:13:42,000
That becomes x squared plus y
squared less than two.

199
00:13:42,000 --> 00:13:52,000
So, that means that's a disk of
radius square root of two,

200
00:13:52,000 --> 00:13:56,000
OK?
So, we kind of knew in advance

201
00:13:56,000 --> 00:14:01,000
it was going to be a disk,
but what we've learned now is

202
00:14:01,000 --> 00:14:05,000
that this radius is square root
of two.

203
00:14:05,000 --> 00:14:08,000
So, if we want to set up,
if we really want to set it up

204
00:14:08,000 --> 00:14:13,000
using dy dx like they started,
then we can do it because we

205
00:14:13,000 --> 00:14:16,000
know,
so, for the middle integral,

206
00:14:16,000 --> 00:14:19,000
now,
we want to fix a value of x.

207
00:14:19,000 --> 00:14:21,000
And, for that fixed value of x,
we want to figure out the

208
00:14:21,000 --> 00:14:25,000
bounds for y.
Well, the answer is y goes from

209
00:14:25,000 --> 00:14:26,000
here to here.
What's here?

210
00:14:26,000 --> 00:14:31,000
Well, here, y is square root of
two minus x squared.

211
00:14:31,000 --> 00:14:36,000
And, here it's negative square
root of two minus x squared.

212
00:14:36,000 --> 00:14:40,000
So, y will go from negative
square root of two minus x

213
00:14:40,000 --> 00:14:46,000
squared to positive square root.
And then, x will go from

214
00:14:46,000 --> 00:14:52,000
negative root two to root two.
OK, if that's not completely

215
00:14:52,000 --> 00:14:55,000
clear to you,
then I encourage you to go over

216
00:14:55,000 --> 00:14:58,000
how we set up double integrals
again.

217
00:14:58,000 --> 00:15:02,000
OK, does that make sense,
kind of?

218
00:15:02,000 --> 00:15:17,000
Yeah?
Well, so, when we set up,

219
00:15:17,000 --> 00:15:20,000
remember, we are setting up a
double integral,

220
00:15:20,000 --> 00:15:23,000
dy dx here.
So, when we do it dy dx,

221
00:15:23,000 --> 00:15:27,000
it means we slice this region
of a plane by vertical line

222
00:15:27,000 --> 00:15:30,000
segments.
So, this middle guy would be

223
00:15:30,000 --> 00:15:33,000
what used to be the inner
integral.

224
00:15:33,000 --> 00:15:36,000
So, in the inner,
remember, you fix the value of

225
00:15:36,000 --> 00:15:39,000
x, and you ask yourself,
what is the range of values of

226
00:15:39,000 --> 00:15:43,000
y in my region?
So, y goes from here to here,

227
00:15:43,000 --> 00:15:46,000
and what here and here are
depends on the value of x.

228
00:15:46,000 --> 00:15:48,000
How?
Well, we have to find the

229
00:15:48,000 --> 00:15:50,000
relation between x and y at
these points.

230
00:15:50,000 --> 00:15:53,000
These points are on the circle
of radius root two.

231
00:15:53,000 --> 00:15:56,000
So, if you want this circle
maybe I should have written,

232
00:15:56,000 --> 00:15:58,000
is x squared plus y squared
equals two.

233
00:15:58,000 --> 00:16:03,000
And, if you solve for y,
given x, you get plus minus

234
00:16:03,000 --> 00:16:07,000
root of two minus x squared,
OK?

235
00:16:07,000 --> 00:16:11,000
Yes?
Is there a way to compute this

236
00:16:11,000 --> 00:16:13,000
with symmetry?
Well, certainly,

237
00:16:13,000 --> 00:16:15,000
yeah, this solid looks
sufficiently symmetric,

238
00:16:15,000 --> 00:16:17,000
but actually you could
certainly,

239
00:16:17,000 --> 00:16:19,000
if you don't want to do the
whole disk,

240
00:16:19,000 --> 00:16:21,000
you could just do quarter
disks,

241
00:16:21,000 --> 00:16:25,000
and multiply by four.
You could even just look at the

242
00:16:25,000 --> 00:16:29,000
lower half of the solid,
and multiply them by two,

243
00:16:29,000 --> 00:16:33,000
so, total by eight.
So, yeah, certainly there's

244
00:16:33,000 --> 00:16:36,000
lots of ways to make it slightly
easier by using symmetry.

245
00:16:36,000 --> 00:16:39,000
Now, the most spectacular way
to use symmetry here,

246
00:16:39,000 --> 00:16:41,000
of course, is to use that we
have this rotation symmetry and

247
00:16:41,000 --> 00:16:45,000
switch,
actually, not do this guy in xy

248
00:16:45,000 --> 00:16:50,000
coordinates but instead in polar
coordinates.

249
00:16:50,000 --> 00:17:08,000
So -- So, the smarter thing to
do would be to use polar

250
00:17:08,000 --> 00:17:21,000
coordinates instead of x and y.
Of course, we want to keep z.

251
00:17:21,000 --> 00:17:23,000
I mean, we are very happy with
z the way it is.

252
00:17:23,000 --> 00:17:28,000
But, we'll just change x and y
to R cos theta,

253
00:17:28,000 --> 00:17:31,000
R sine theta,
OK, because,

254
00:17:31,000 --> 00:17:37,000
well, let's see actually how we
would evaluate this guy.

255
00:17:37,000 --> 00:17:46,000
So, well actually, let's not.
It's kind of boring.

256
00:17:46,000 --> 00:17:50,000
So, let me just point out one
small thing here,

257
00:17:50,000 --> 00:17:54,000
sorry, before I do that.
So, if you start computing the

258
00:17:54,000 --> 00:17:59,000
inner integral,
OK, so let me not do that yet,

259
00:17:59,000 --> 00:18:03,000
sorry,
so if you try to compute the

260
00:18:03,000 --> 00:18:06,000
inner integral,
you'll be integrating from x

261
00:18:06,000 --> 00:18:11,000
squared plus y squared to four
minus x squared minus y squared

262
00:18:11,000 --> 00:18:15,000
dz.
Well, that will integrate to z

263
00:18:15,000 --> 00:18:22,000
between these two bounds.
So, you will get four minus two

264
00:18:22,000 --> 00:18:27,000
x squared minus two y squared.
Now, when you put that into the

265
00:18:27,000 --> 00:18:33,000
remaining ones,
you'll get something that's

266
00:18:33,000 --> 00:18:41,000
probably not very pleasant of
four minus two x squared minus

267
00:18:41,000 --> 00:18:49,000
two y squared dy dx.
And here, you see that to

268
00:18:49,000 --> 00:18:56,000
evaluate this,
you would switch to polar

269
00:18:56,000 --> 00:18:59,000
coordinates.
Oh, by the way,

270
00:18:59,000 --> 00:19:04,000
so if your initial instincts
had been to,

271
00:19:04,000 --> 00:19:06,000
given that you just want the
volume,

272
00:19:06,000 --> 00:19:09,000
you could also have found the
volume just by doing a double

273
00:19:09,000 --> 00:19:12,000
integral of the height between
the top and bottom.

274
00:19:12,000 --> 00:19:14,000
Well, you would just have
gotten this, right,

275
00:19:14,000 --> 00:19:17,000
because this is the height
between top and bottom.

276
00:19:17,000 --> 00:19:21,000
So, it's all the same.
It doesn't really matter.

277
00:19:21,000 --> 00:19:23,000
But with this,
of course, we will be able to

278
00:19:23,000 --> 00:19:26,000
integrate all sorts of
functions, not just one over the

279
00:19:26,000 --> 00:19:31,000
solid.
So, we will be able to do much

280
00:19:31,000 --> 00:19:35,000
more than just volumes.
OK, so let's see,

281
00:19:35,000 --> 00:19:37,000
how do we do it with polar
coordinates instead?

282
00:19:37,000 --> 00:19:53,000
Well, so -- Well,
that would become,

283
00:19:53,000 --> 00:20:02,000
so let's see.
So, I want to keep dz.

284
00:20:02,000 --> 00:20:10,000
But then, dx dy or dy dx would
become r dr d theta.

285
00:20:10,000 --> 00:20:13,000
And, if I try to set up the
bounds, well,

286
00:20:13,000 --> 00:20:17,000
I probably shouldn't keep this
x squared plus y squared around.

287
00:20:17,000 --> 00:20:20,000
But, x squared plus y squared
is easy in terms of r and theta.

288
00:20:20,000 --> 00:20:26,000
That's just r squared.
OK, I mean, in general I could

289
00:20:26,000 --> 00:20:28,000
have something that depends also
on theta.

290
00:20:28,000 --> 00:20:32,000
That's perfectly legitimate.
But here, it simplifies,

291
00:20:32,000 --> 00:20:36,000
and this guy up here,
four minus x squared minus y

292
00:20:36,000 --> 00:20:40,000
squared becomes four minus r
squared.

293
00:20:40,000 --> 00:20:43,000
And now, the integral that we
have to do over r and theta,

294
00:20:43,000 --> 00:20:45,000
well, we look again at the
shadow.

295
00:20:45,000 --> 00:20:48,000
The shadow is still a disk of
radius root two.

296
00:20:48,000 --> 00:20:52,000
That hasn't changed.
And now, we know how to set up

297
00:20:52,000 --> 00:20:54,000
this integral in polar
coordinates.

298
00:20:54,000 --> 00:21:01,000
r goes from zero to root two,
and theta goes from zero to two

299
00:21:01,000 --> 00:21:11,000
pi.
OK, and now it becomes actually

300
00:21:11,000 --> 00:21:20,000
easier to evaluate.
OK, so now we have actually a

301
00:21:20,000 --> 00:21:24,000
name for this because we're
doing it in space.

302
00:21:24,000 --> 00:21:28,000
So, these are called,
actually, cylindrical

303
00:21:28,000 --> 00:21:30,000
coordinates.
So, in fact,

304
00:21:30,000 --> 00:21:35,000
you already knew about
cylindrical coordinates even if

305
00:21:35,000 --> 00:21:40,000
you did not know the name.
OK, so the idea of cylindrical

306
00:21:40,000 --> 00:21:45,000
coordinates is that instead of
x, y, and z, to locate a point

307
00:21:45,000 --> 00:21:48,000
in space, you will use three
coordinates.

308
00:21:48,000 --> 00:22:00,000
One of them is basically how
high it is above the xy plane.

309
00:22:00,000 --> 00:22:04,000
So, that will be z.
And then, you will use polar

310
00:22:04,000 --> 00:22:08,000
coordinates for the projection
of your point on the xy plane.

311
00:22:08,000 --> 00:22:12,000
So, r will be the distance from
the z axis.

312
00:22:12,000 --> 00:22:17,000
And theta will be the angle
from the x axis

313
00:22:17,000 --> 00:22:21,000
counterclockwise.
So, the one thing to be careful

314
00:22:21,000 --> 00:22:24,000
about is because of the usual
convention, that we make the x

315
00:22:24,000 --> 00:22:27,000
axis point toward us.
Theta equals zero is no longer

316
00:22:27,000 --> 00:22:30,000
to the right.
Now, theta equals zero is to

317
00:22:30,000 --> 00:22:34,000
the front, and the angel is
measured from the front

318
00:22:34,000 --> 00:22:39,000
counterclockwise.
OK, so,

319
00:22:39,000 --> 00:22:41,000
and of course,
if you want to know how to

320
00:22:41,000 --> 00:22:44,000
convert between x,
y, z and r theta z,

321
00:22:44,000 --> 00:22:49,000
well, the formulas are just the
same as in usual polar

322
00:22:49,000 --> 00:22:52,000
coordinates.
R cos theta,

323
00:22:52,000 --> 00:22:56,000
r sine theta,
and z remain z.

324
00:22:56,000 --> 00:22:59,000
OK, so why are these called
cylindrical coordinates,

325
00:22:59,000 --> 00:23:02,000
by the way?
Well, let's say that I gave you

326
00:23:02,000 --> 00:23:07,000
the equation r equals a,
where a is some constant.

327
00:23:07,000 --> 00:23:12,000
Say r equals one, for example.
So, r equals one in 2D,

328
00:23:12,000 --> 00:23:15,000
that used to be just a circle
of radius one.

329
00:23:15,000 --> 00:23:19,000
Now, in space,
a single equation actually

330
00:23:19,000 --> 00:23:23,000
defines a surface,
not just a curve anymore.

331
00:23:23,000 --> 00:23:26,000
And, the set of points where r
is a, well, that's all the

332
00:23:26,000 --> 00:23:29,000
points that are distance a from
the z axis.

333
00:23:29,000 --> 00:23:34,000
So, in fact,
what you get this way is a

334
00:23:34,000 --> 00:23:41,000
cylinder of radius a centered on
the z axis.

335
00:23:41,000 --> 00:23:48,000
OK, so that's why they are
called cylindrical coordinates.

336
00:23:48,000 --> 00:23:51,000
By the way, so now,
similarly, if you look at the

337
00:23:51,000 --> 00:23:55,000
equation theta equals some given
value, well, so that used to be

338
00:23:55,000 --> 00:23:59,000
just a ray from the origin.
Now, that becomes a vertical

339
00:23:59,000 --> 00:24:01,000
half plane.
For example,

340
00:24:01,000 --> 00:24:04,000
if I set the value of theta and
let r and z vary,

341
00:24:04,000 --> 00:24:08,000
well, r is always positive,
but basically that means I am

342
00:24:08,000 --> 00:24:13,000
taking a vertical plane that
comes out in this direction.

343
00:24:13,000 --> 00:24:18,000
OK, any questions about
cylindrical coordinates?

344
00:24:18,000 --> 00:24:27,000
Yes?
Yeah, so I'm saying when you

345
00:24:27,000 --> 00:24:30,000
fix theta, you get only a half
plane, not a full plane.

346
00:24:30,000 --> 00:24:33,000
I mean, it goes all the way up
and down, but it doesn't go back

347
00:24:33,000 --> 00:24:35,000
to the other side of the z axis.
Why?

348
00:24:35,000 --> 00:24:39,000
That's because r is always
positive by convention.

349
00:24:39,000 --> 00:24:41,000
So, for example,
here, we say theta is zero.

350
00:24:41,000 --> 00:24:44,000
At the back, we say theta is pi.
We don't say theta is zero and

351
00:24:44,000 --> 00:24:47,000
r is negative.
We say r is positive and theta

352
00:24:47,000 --> 00:24:50,000
is pi.
It's a convention, largely.

353
00:24:50,000 --> 00:24:52,000
But, sticking with this
convention really will help you

354
00:24:52,000 --> 00:24:54,000
to set up the integrals
properly.

355
00:24:54,000 --> 00:24:58,000
I mean, otherwise there is just
too much risk for mistakes.

356
00:24:58,000 --> 00:25:08,000
Yes?
Well, so the question is if I

357
00:25:08,000 --> 00:25:11,000
were to use symmetry to do this
one, would I multiply by four or

358
00:25:11,000 --> 00:25:13,000
by two?
Well, it depends on how much

359
00:25:13,000 --> 00:25:16,000
symmetry you are using.
So, I mean, it's your choice.

360
00:25:16,000 --> 00:25:19,000
You can multiply by two,
by four, by eight depending on

361
00:25:19,000 --> 00:25:22,000
how much you cut it.
So, it depends on what symmetry

362
00:25:22,000 --> 00:25:24,000
you use, if you use symmetry
between top and bottom you'd

363
00:25:24,000 --> 00:25:27,000
say, well, the volume is twice
the lower half.

364
00:25:27,000 --> 00:25:30,000
If you use the left and right
half, you would say it's twice

365
00:25:30,000 --> 00:25:34,000
each half.
If you cut it into four pieces,

366
00:25:34,000 --> 00:25:37,000
and so on.
So, and again,

367
00:25:37,000 --> 00:25:41,000
you don't have to use the
symmetry.

368
00:25:41,000 --> 00:25:44,000
If you don't think of using
polar coordinates,

369
00:25:44,000 --> 00:25:46,000
then it can save you from
doing,

370
00:25:46,000 --> 00:25:47,000
you know, you can just start at
zero here and here,

371
00:25:47,000 --> 00:26:01,000
and simplify things a tiny bit.
But, OK, yes?

372
00:26:01,000 --> 00:26:04,000
So, to define a vertical full
plane, well, first of all it

373
00:26:04,000 --> 00:26:07,000
depends on whether it passes
through the z axis or not.

374
00:26:07,000 --> 00:26:09,000
If it doesn't,
then you'd have to remember how

375
00:26:09,000 --> 00:26:13,000
you do in polar coordinates.
I mean, basically the answer

376
00:26:13,000 --> 00:26:16,000
is, if you have a vertical
plane, so, it doesn't depend on

377
00:26:16,000 --> 00:26:18,000
z.
The equation does not involve z.

378
00:26:18,000 --> 00:26:21,000
It only involves r and theta.
And, how it involves r and

379
00:26:21,000 --> 00:26:24,000
theta is exactly the same as
when you do a line in polar

380
00:26:24,000 --> 00:26:27,000
coordinates in the plane.
So, if it's a line passing

381
00:26:27,000 --> 00:26:29,000
through the origin,
you say, well,

382
00:26:29,000 --> 00:26:32,000
theta is either some value or
the other one.

383
00:26:32,000 --> 00:26:33,000
If it's a line that doesn't
passes to the origin,

384
00:26:33,000 --> 00:26:38,000
but it's more tricky.
But hopefully you've seen how

385
00:26:38,000 --> 00:26:49,000
to do that.
OK, let's move on a bit.

386
00:26:49,000 --> 00:26:53,000
So, one thing to know,
I mean, basically,

387
00:26:53,000 --> 00:26:57,000
the important thing to remember
is that the volume element in

388
00:26:57,000 --> 00:27:05,000
cylindrical coordinates,
well, dx dy dz becomes r dr d

389
00:27:05,000 --> 00:27:08,000
theta dz.
And, that shouldn't be

390
00:27:08,000 --> 00:27:12,000
surprising because that's just
dx dy becomes r dr d theta.

391
00:27:12,000 --> 00:27:18,000
And, dz remains dz.
I mean, so, the way to think

392
00:27:18,000 --> 00:27:19,000
about it,
if you want,

393
00:27:19,000 --> 00:27:25,000
is that if you take a little
piece of solid in space,

394
00:27:25,000 --> 00:27:31,000
so it has some height, delta z,
and it has a base which has

395
00:27:31,000 --> 00:27:36,000
some area delta A,
then the small volume, delta v,

396
00:27:36,000 --> 00:27:41,000
is equal to the area of a base
times the height.

397
00:27:41,000 --> 00:27:43,000
So, now, when you make the
things infinitely small,

398
00:27:43,000 --> 00:27:51,000
you will get dV is dA times dz,
and you can use whichever

399
00:27:51,000 --> 00:27:56,000
formula you want for area in the
xy plane.

400
00:27:56,000 --> 00:28:00,000
OK, now in practice,
you choose which order you

401
00:28:00,000 --> 00:28:03,000
integrate in.
As you have probably seen,

402
00:28:03,000 --> 00:28:07,000
a favorite of mine is z first
because very often you'll know

403
00:28:07,000 --> 00:28:10,000
what the top and bottom of your
solid look like,

404
00:28:10,000 --> 00:28:13,000
and then you will reduce to
just something in the xy plane.

405
00:28:13,000 --> 00:28:18,000
But, there might be situations
where it's actually easier to

406
00:28:18,000 --> 00:28:22,000
start first with dx dy or r dr d
theta, and then save dz for

407
00:28:22,000 --> 00:28:24,000
last.
I mean, if you seen how to,

408
00:28:24,000 --> 00:28:27,000
in single variable calculus,
the disk and shell methods for

409
00:28:27,000 --> 00:28:30,000
finding volumes,
that's exactly the dilemma of

410
00:28:30,000 --> 00:28:39,000
shells versus disks.
One of them is you do z first.

411
00:28:39,000 --> 00:28:49,000
The other is you do z last.
OK, so what are things we can

412
00:28:49,000 --> 00:28:56,000
do now with triple integrals?
Well, we can find the volume of

413
00:28:56,000 --> 00:29:00,000
solids by just integrating dV.
And, we've seen that.

414
00:29:00,000 --> 00:29:06,000
We can find the mass of a solid.
OK, so if we have a density,

415
00:29:06,000 --> 00:29:10,000
delta, which,
remember, delta is basically

416
00:29:10,000 --> 00:29:17,000
the mass divided by the volume.
OK, so the small mass element,

417
00:29:17,000 --> 00:29:23,000
maybe I should have written
that as dm, the mass element,

418
00:29:23,000 --> 00:29:27,000
is density times dV.
So now, this is the real

419
00:29:27,000 --> 00:29:29,000
physical density.
If you are given a material,

420
00:29:29,000 --> 00:29:33,000
usually, the density will be in
grams per cubic meter or cubic

421
00:29:33,000 --> 00:29:35,000
inch, or whatever.
I mean, there is tons of

422
00:29:35,000 --> 00:29:37,000
different units.
But, so then,

423
00:29:37,000 --> 00:29:43,000
the mass of your solid will be
just the triple integral of

424
00:29:43,000 --> 00:29:50,000
density, dV because you just sum
the mass of each little piece.

425
00:29:50,000 --> 00:29:53,000
And, of course,
if the density is one,

426
00:29:53,000 --> 00:29:55,000
then it just becomes the
volume.

427
00:29:55,000 --> 00:29:59,000
OK,
now, it shouldn't be surprising

428
00:29:59,000 --> 00:30:02,000
to you that we can also do
classics that we had seen in the

429
00:30:02,000 --> 00:30:05,000
plane such as the average value
of a function,

430
00:30:05,000 --> 00:30:07,000
the center of mass,
and moment of inertia.

431
00:30:38,000 --> 00:30:47,000
OK, so the average value of the
function f of x,

432
00:30:47,000 --> 00:30:51,000
y, z in the region,
r,

433
00:30:51,000 --> 00:30:57,000
that would be f bar,
would be one over the volume of

434
00:30:57,000 --> 00:31:02,000
the region times the triple
integral of f dV.

435
00:31:02,000 --> 00:31:09,000
Or, if we have a density,
and we want to take a weighted

436
00:31:09,000 --> 00:31:21,000
average -- Then we take one over
the mass where the mass is the

437
00:31:21,000 --> 00:31:32,000
triple integral of the density
times the triple integral of f

438
00:31:32,000 --> 00:31:36,000
density dV.
So, as particular cases,

439
00:31:36,000 --> 00:31:39,000
there is, again,
the notion of center of mass of

440
00:31:39,000 --> 00:31:41,000
the solid.
So, that's the point that

441
00:31:41,000 --> 00:31:44,000
somehow right in the middle of
the solid.

442
00:31:44,000 --> 00:31:48,000
That's the point mass by which
there is a point at which you

443
00:31:48,000 --> 00:31:53,000
should put point mass so that it
would be equivalent from the

444
00:31:53,000 --> 00:31:57,000
point of view of dealing with
forces and translation effects,

445
00:31:57,000 --> 00:32:03,000
of course, not for rotation.
But, so the center of mass of a

446
00:32:03,000 --> 00:32:09,000
solid is just given by taking
the average values of x,

447
00:32:09,000 --> 00:32:14,000
y, and z.
OK, so there is a special case

448
00:32:14,000 --> 00:32:21,000
where, so, x bar is one over the
mass times triple integral of x

449
00:32:21,000 --> 00:32:31,000
density dV.
And, same thing with y and z.

450
00:32:31,000 --> 00:32:33,000
And, of course,
very often, you can use

451
00:32:33,000 --> 00:32:37,000
symmetry to not have to compute
all three of them.

452
00:32:37,000 --> 00:32:39,000
For example,
if you look at this solid that

453
00:32:39,000 --> 00:32:41,000
we had, well,
I guess I've erased it now.

454
00:32:41,000 --> 00:32:43,000
But, if you remember what it
looked, well,

455
00:32:43,000 --> 00:32:45,000
it was pretty obvious that the
center of mass would be in the z

456
00:32:45,000 --> 00:32:47,000
axis.
So, no need to waste time

457
00:32:47,000 --> 00:32:49,000
considering x bar and y bar.

458
00:33:03,000 --> 00:33:08,000
And, in fact,
you can also find z bar by

459
00:33:08,000 --> 00:33:16,000
symmetry between the top and
bottom, and let you figure that

460
00:33:16,000 --> 00:33:18,000
out.
Of course, symmetry only works,

461
00:33:18,000 --> 00:33:20,000
I should say,
symmetry only works if the

462
00:33:20,000 --> 00:33:25,000
density is also symmetric.
If I had taken my guy to be

463
00:33:25,000 --> 00:33:31,000
heavier at the front than at the
back, then it would no longer be

464
00:33:31,000 --> 00:33:37,000
true that x bar would be zero.
OK, next on the list is moment

465
00:33:37,000 --> 00:33:40,000
of inertia.
Actually, in a way,

466
00:33:40,000 --> 00:33:45,000
moment of inertia in 3D is
easier conceptually than in 2D.

467
00:33:45,000 --> 00:33:49,000
So, why is that?
Well, because now the various

468
00:33:49,000 --> 00:33:53,000
flavors that we had come
together in a nice way.

469
00:33:53,000 --> 00:33:56,000
So, the moment of inertia of an
axis,

470
00:33:56,000 --> 00:33:58,000
sorry, with respect to an axis
would be,

471
00:33:58,000 --> 00:34:06,000
again, given by the triple
integral of the distance to the

472
00:34:06,000 --> 00:34:11,000
axis squared times density,
times dV.

473
00:34:11,000 --> 00:34:15,000
And, in particular,
we have our solid.

474
00:34:15,000 --> 00:34:19,000
And, we might skewer it using
any of the coordinate axes and

475
00:34:19,000 --> 00:34:22,000
then try to rotate it about one
of the axes.

476
00:34:22,000 --> 00:34:24,000
So, we have three different
possibilities,

477
00:34:24,000 --> 00:34:27,000
of course, the x,
y, or z axis.

478
00:34:27,000 --> 00:34:29,000
And, so now,
rotating about the z axis

479
00:34:29,000 --> 00:34:34,000
actually corresponds to when we
were just doing things for flat

480
00:34:34,000 --> 00:34:38,000
objects in the xy plane.
That corresponded to rotating

481
00:34:38,000 --> 00:34:40,000
about the origin.
So, secretly,

482
00:34:40,000 --> 00:34:42,000
we were saying we were rotating
about the point.

483
00:34:42,000 --> 00:34:44,000
But actually,
it was just rotating about the

484
00:34:44,000 --> 00:34:47,000
z axis.
Just I didn't want to introduce

485
00:34:47,000 --> 00:34:51,000
the z coordinate that we didn't
actually need at the time.

486
00:35:18,000 --> 00:35:23,000
So -- [APPLAUSE]
OK, so moment of inertia about

487
00:35:23,000 --> 00:35:28,000
the z axis, so,
what's the distance to the z

488
00:35:28,000 --> 00:35:31,000
axis?
Well, we've said that's exactly

489
00:35:31,000 --> 00:35:34,000
r.
That's the cylindrical

490
00:35:34,000 --> 00:35:39,000
coordinate, r.
So, the square of a distance is

491
00:35:39,000 --> 00:35:44,000
just r squared.
Now, if you didn't want to do

492
00:35:44,000 --> 00:35:49,000
it in cylindrical coordinates
then, of course,

493
00:35:49,000 --> 00:35:55,000
r squared is just x squared
plus y squared.

494
00:35:55,000 --> 00:35:58,000
Square of distance from the z
axis is just x squared plus y

495
00:35:58,000 --> 00:36:00,000
squared.
Similarly, now,

496
00:36:00,000 --> 00:36:04,000
if you want the distance from
the x axis, well,

497
00:36:04,000 --> 00:36:07,000
that will be y squared plus z
squared.

498
00:36:07,000 --> 00:36:09,000
OK, try to convince yourselves
of the picture,

499
00:36:09,000 --> 00:36:13,000
or else just argue by symmetry:
you know, if you change the

500
00:36:13,000 --> 00:36:18,000
positions of the axis.
So, moment of inertia about the

501
00:36:18,000 --> 00:36:25,000
x axis is the double integral of
y squared plus z squared delta

502
00:36:25,000 --> 00:36:29,000
dV.
And moment of inertia about the

503
00:36:29,000 --> 00:36:34,000
y axis is the same thing,
but now with x squared plus z

504
00:36:34,000 --> 00:36:36,000
squared.
And so, now,

505
00:36:36,000 --> 00:36:39,000
if you try to apply these
things for flat solids that are

506
00:36:39,000 --> 00:36:42,000
in the xy plane,
so where there's no z to look

507
00:36:42,000 --> 00:36:45,000
at,
well, you see these formulas

508
00:36:45,000 --> 00:36:48,000
become the old formulas that we
had.

509
00:36:48,000 --> 00:36:56,000
But now, they all fit together
in a more symmetric way.

510
00:36:56,000 --> 00:37:04,000
OK, any questions about that?
No?

511
00:37:04,000 --> 00:37:08,000
OK, so these are just formulas
to remember.

512
00:37:08,000 --> 00:37:21,000
So, OK, let's do an example.
Was there a question that I

513
00:37:21,000 --> 00:37:24,000
missed?
No?

514
00:37:24,000 --> 00:37:34,000
OK, so let's find the moment of
inertia about the z axis of a

515
00:37:34,000 --> 00:37:44,000
solid cone -- -- between z
equals a times r and z equals b.

516
00:37:44,000 --> 00:37:47,000
So, just to convince you that
it's a cone, so,

517
00:37:47,000 --> 00:37:52,000
z equals a times r means the
height is proportional to the

518
00:37:52,000 --> 00:37:57,000
distance from the z axis.
So, let's look at what we get

519
00:37:57,000 --> 00:38:00,000
if we just do it in the plane of
a blackboard.

520
00:38:00,000 --> 00:38:04,000
So, if I go to the right here,
r is just the distance from the

521
00:38:04,000 --> 00:38:06,000
x axis.
The height should be

522
00:38:06,000 --> 00:38:09,000
proportional with
proportionality factor A.

523
00:38:09,000 --> 00:38:14,000
So, that means I take a line
with slope A.

524
00:38:14,000 --> 00:38:16,000
If I'm on the left,
well, it's the same story

525
00:38:16,000 --> 00:38:19,000
except distance to the z axis is
still positive.

526
00:38:19,000 --> 00:38:22,000
So, I get the symmetric thing.
And, in fact,

527
00:38:22,000 --> 00:38:25,000
it doesn't matter which
vertical plane I do it in.

528
00:38:25,000 --> 00:38:28,000
This is the same if I rotate
about.

529
00:38:28,000 --> 00:38:32,000
See, there's no theta in here.
So, it's the same in all

530
00:38:32,000 --> 00:38:36,000
directions.
So, I claim it's a cone where

531
00:38:36,000 --> 00:38:43,000
the slope of the rays is A.
OK, and z equals b.

532
00:38:43,000 --> 00:38:50,000
Well, that just means we stop
in our horizontal plane at

533
00:38:50,000 --> 00:38:53,000
height b.
OK, so that's solid cone really

534
00:38:53,000 --> 00:38:58,000
just looks like this.
That's our solid.

535
00:38:58,000 --> 00:39:02,000
OK, so it has a flat top,
that circular top,

536
00:39:02,000 --> 00:39:08,000
and then the point is at v.
The tip of it is at the origin.

537
00:39:08,000 --> 00:39:12,000
So, let's try to compute its
moment of inertia about the z

538
00:39:12,000 --> 00:39:14,000
axis.
So, that means maybe this is

539
00:39:14,000 --> 00:39:16,000
like the top that you are going
to spin.

540
00:39:16,000 --> 00:39:21,000
And, it tells you how hard it
is to actually spin that top.

541
00:39:21,000 --> 00:39:24,000
Actually, that's also useful if
you're going to do mechanical

542
00:39:24,000 --> 00:39:27,000
engineering because if you are
trying to design gears,

543
00:39:27,000 --> 00:39:28,000
and things like that that will
rotate,

544
00:39:28,000 --> 00:39:31,000
you might want to know exactly
how much effort you'll have to

545
00:39:31,000 --> 00:39:33,000
put to actually get them to
spin,

546
00:39:33,000 --> 00:39:37,000
and whether you're actually
going to have a strong enough

547
00:39:37,000 --> 00:39:39,000
engine, or whatever,
to do it.

548
00:39:39,000 --> 00:39:41,000
OK, so what's the moment of
inertia of this guy?

549
00:39:41,000 --> 00:39:44,000
Well, that's the triple
integral of, well,

550
00:39:44,000 --> 00:39:49,000
we have to choose x squared
plus y squared or r squared.

551
00:39:49,000 --> 00:39:52,000
Let's see, I think I want to
use cylindrical coordinates to

552
00:39:52,000 --> 00:39:57,000
do that, given the shape.
So, we use r squared.

553
00:39:57,000 --> 00:40:02,000
I might have a density that
let's say the density is one.

554
00:40:02,000 --> 00:40:06,000
So, I don't have density.
I still have dV.

555
00:40:06,000 --> 00:40:13,000
Now, it will be my choice to
choose between doing the dz

556
00:40:13,000 --> 00:40:17,000
first or doing r dr d theta
first.

557
00:40:17,000 --> 00:40:20,000
Just to show you how it goes
the other way around,

558
00:40:20,000 --> 00:40:23,000
let me do it r dr d theta dz
this time.

559
00:40:23,000 --> 00:40:29,000
Then you can decide on a
case-by-case basis which one you

560
00:40:29,000 --> 00:40:33,000
like best.
OK, so if we do it in this

561
00:40:33,000 --> 00:40:36,000
direction, it means that in the
inner and middle integrals,

562
00:40:36,000 --> 00:40:40,000
we've fixed a value of z.
And, for that particular value

563
00:40:40,000 --> 00:40:45,000
of z, we'll be actually slicing
our solid by a horizontal plane,

564
00:40:45,000 --> 00:40:47,000
and looking at what we get,
OK?

565
00:40:47,000 --> 00:40:54,000
So, what does that look like?
Well, I fixed a value of z,

566
00:40:54,000 --> 00:40:59,000
and I slice my solid by a
horizontal plane.

567
00:40:59,000 --> 00:41:04,000
Well, I'm going to get a circle
certainly.

568
00:41:04,000 --> 00:41:07,000
What's the radius,
well, a disk actually,

569
00:41:07,000 --> 00:41:11,000
what's the radius of the disk?
Yeah, the radius of the disk

570
00:41:11,000 --> 00:41:14,000
should be z over a because the
equation of that cone,

571
00:41:14,000 --> 00:41:19,000
we said it's z equals ar.
So, if you flip it around,

572
00:41:19,000 --> 00:41:24,000
so, maybe I should switch to
another blackboard.

573
00:41:24,000 --> 00:41:33,000
So, the equation of a cone is z
equals ar, or equivalently r

574
00:41:33,000 --> 00:41:40,000
equals z over a.
So, for a given value of z,

575
00:41:40,000 --> 00:41:49,000
I will get, this guy will be a
disk of radius z over a.

576
00:41:49,000 --> 00:41:55,000
OK, so, moment of inertia is
going to be, well,

577
00:41:55,000 --> 00:41:59,000
we said r squared,
r dr d theta dz.

578
00:41:59,000 --> 00:42:02,000
Now, so, to set up the inner
and middle integrals,

579
00:42:02,000 --> 00:42:06,000
I just set up a double integral
over this disk of radius z over

580
00:42:06,000 --> 00:42:07,000
a.
So, it's easy.

581
00:42:07,000 --> 00:42:14,000
r goes from zero to z over a.
Theta goes from zero to 2pi.

582
00:42:14,000 --> 00:42:17,000
OK, and then,
well, if I set up the bounds

583
00:42:17,000 --> 00:42:20,000
for z, now it's my outer
variable.

584
00:42:20,000 --> 00:42:24,000
So, the question I have to ask
is what is the first slice?

585
00:42:24,000 --> 00:42:28,000
What is the last slice?
So, the bottommost value of z

586
00:42:28,000 --> 00:42:33,000
would be zero,
and the topmost would be b.

587
00:42:33,000 --> 00:42:37,000
And so, that's it I get.
So, exercise,

588
00:42:37,000 --> 00:42:42,000
it's not very hard.
Try to set it up the other way

589
00:42:42,000 --> 00:42:47,000
around with dz first and then r
dr d theta.

590
00:42:47,000 --> 00:42:49,000
It's pretty much the same level
of difficulty.

591
00:42:49,000 --> 00:42:52,000
I'm sure you can do both of
them.

592
00:42:52,000 --> 00:42:57,000
So, and also,
if you want to practice

593
00:42:57,000 --> 00:43:03,000
calculations,
you should end up getting pi b

594
00:43:03,000 --> 00:43:10,000
to the five over 10a to the four
if I got it right.

595
00:43:10,000 --> 00:43:14,000
OK, let me finish with one more
example.

596
00:43:14,000 --> 00:43:17,000
I'm trying to give you plenty
of practice because in case you

597
00:43:17,000 --> 00:43:19,000
haven't noticed,
Monday is a holiday.

598
00:43:19,000 --> 00:43:22,000
So, you don't have recitation
on Monday, which is good.

599
00:43:22,000 --> 00:43:25,000
But it means that there will be
lots of stuff to cover on

600
00:43:25,000 --> 00:43:47,000
Wednesday.
So -- Thank you.

601
00:43:47,000 --> 00:43:58,000
OK, so third example,
let's say that I want to just

602
00:43:58,000 --> 00:44:09,000
set up a triple integral for the
region where z is bigger than

603
00:44:09,000 --> 00:44:18,000
one minus y inside the unit ball
centered at the origin.

604
00:44:18,000 --> 00:44:24,000
So, the unit ball is just,
you know, well,

605
00:44:24,000 --> 00:44:30,000
stay inside of the unit sphere.
So, its equation,

606
00:44:30,000 --> 00:44:33,000
if you want,
would be x squared plus y

607
00:44:33,000 --> 00:44:35,000
squared plus z squared less than
one.

608
00:44:35,000 --> 00:44:37,000
OK, so that's one thing you
should remember.

609
00:44:37,000 --> 00:44:40,000
The equation of a sphere
centered at the origin is x

610
00:44:40,000 --> 00:44:44,000
squared plus y squared plus z
squared equals radius squared.

611
00:44:44,000 --> 00:44:48,000
And now, we are going to take
this plane, z equals one minus

612
00:44:48,000 --> 00:44:50,000
y.
So, if you think about it,

613
00:44:50,000 --> 00:44:52,000
it's parallel to the x axis
because there's no x in its

614
00:44:52,000 --> 00:44:55,000
coordinate in its equation.
At the origin,

615
00:44:55,000 --> 00:44:59,000
the height is one.
So, it starts right here at one.

616
00:44:59,000 --> 00:45:04,000
And, it slopes down with y with
slope one.

617
00:45:04,000 --> 00:45:07,000
OK, so it's a plane that comes
straight out here,

618
00:45:07,000 --> 00:45:10,000
and it intersects the sphere,
so here and here,

619
00:45:10,000 --> 00:45:13,000
but also at other points in
between.

620
00:45:13,000 --> 00:45:18,000
Any idea what kind of shape
this is?

621
00:45:18,000 --> 00:45:20,000
Well, it's an ellipse,
but it's even more than that.

622
00:45:20,000 --> 00:45:23,000
It's also a circle.
If you slice a sphere by a

623
00:45:23,000 --> 00:45:25,000
plane, you always get a circle.
But, of course,

624
00:45:25,000 --> 00:45:28,000
it's a slanted circle.
So, if you look at it in the xy

625
00:45:28,000 --> 00:45:31,000
plane, if you project it to the
xy plane, that you will get an

626
00:45:31,000 --> 00:45:35,000
ellipse.
OK, so we want to look at this

627
00:45:35,000 --> 00:45:38,000
guy in here.
So, how do we do that?

628
00:45:38,000 --> 00:45:42,000
Well, so maybe I should
actually draw quickly a picture.

629
00:45:42,000 --> 00:45:47,000
So, in the yz plane,
it looks just like this,

630
00:45:47,000 --> 00:45:51,000
OK?
But, if I look at it from above

631
00:45:51,000 --> 00:45:54,000
in the xy plane,
then its shadow,

632
00:45:54,000 --> 00:45:59,000
well, see, it will sit entirely
where y is positive.

633
00:45:59,000 --> 00:46:02,000
So, it sits entirely above
here, and it goes through here

634
00:46:02,000 --> 00:46:04,000
and here.
And, in fact,

635
00:46:04,000 --> 00:46:08,000
when you project that slanted
circle, now you will get an

636
00:46:08,000 --> 00:46:12,000
ellipse.
And, well, I don't really know

637
00:46:12,000 --> 00:46:20,000
how to draw it well,
but it should be something like

638
00:46:20,000 --> 00:46:24,000
this.
OK, so now if you want to try

639
00:46:24,000 --> 00:46:29,000
to set up that double integral,
sorry, the triple integral,

640
00:46:29,000 --> 00:46:37,000
well, so let's say we do it in
rectangular coordinates because

641
00:46:37,000 --> 00:46:41,000
we are really evil.
[LAUGHTER]

642
00:46:41,000 --> 00:46:43,000
So then, the bottom surface,
OK, so we do it with z first.

643
00:46:43,000 --> 00:46:46,000
So, the bottom surface is the
slanted plane.

644
00:46:46,000 --> 00:46:51,000
So, the bottom value would be z
equals one minus y.

645
00:46:51,000 --> 00:46:56,000
The top value is on the sphere.
So, the sphere corresponds to z

646
00:46:56,000 --> 00:47:01,000
equals square root of one minus
x squared minus y squared.

647
00:47:01,000 --> 00:47:05,000
So, you'd go from the plane to
the sphere.

648
00:47:05,000 --> 00:47:09,000
And then, to find the bounds
for x and y, you have to figure

649
00:47:09,000 --> 00:47:13,000
out what exactly,
what the heck is this region

650
00:47:13,000 --> 00:47:15,000
here?
So, what is this region?

651
00:47:15,000 --> 00:47:19,000
Well, we have to figure out,
for what values of x and y the

652
00:47:19,000 --> 00:47:23,000
plane is below the ellipse.
So, the condition is that,

653
00:47:23,000 --> 00:47:25,000
sorry, the plane is below the
sphere.

654
00:47:25,000 --> 00:47:31,000
OK, so, that's when the plane
is below the sphere.

655
00:47:31,000 --> 00:47:37,000
That means one minus y is less
than square root of one minus x

656
00:47:37,000 --> 00:47:41,000
squared minus y squared.
So, you have to somehow

657
00:47:41,000 --> 00:47:43,000
manipulate this to extract
something simpler.

658
00:47:43,000 --> 00:47:47,000
Well, probably the only way to
do it is to square both sides,

659
00:47:47,000 --> 00:47:51,000
one minus y squared should be
less than one minus x squared

660
00:47:51,000 --> 00:47:55,000
minus y squared.
And, if you work hard enough,

661
00:47:55,000 --> 00:47:57,000
you'll find quite an ugly
equation.

662
00:47:57,000 --> 00:48:00,000
But, you can figure out what
are, then, the bounds for x

663
00:48:00,000 --> 00:48:03,000
given y, and then set up the
integral?

664
00:48:03,000 --> 00:48:06,000
So, just to give you a hint,
the bounds on y will be zero to

665
00:48:06,000 --> 00:48:09,000
one.
The bounds on x,

666
00:48:09,000 --> 00:48:10,000
well, I'm not sure you want to
see them,

667
00:48:10,000 --> 00:48:14,000
but in case you do,
it will be from negative square

668
00:48:14,000 --> 00:48:18,000
root of 2y minus 2y squared to
square root of 2y minus 2y

669
00:48:18,000 --> 00:48:20,000
squared.
So, exercise,

670
00:48:20,000 --> 00:48:25,000
figure out how I got these by
starting from that.

671
00:48:25,000 --> 00:48:27,000
Now, of course,
if we just wanted the volume of

672
00:48:27,000 --> 00:48:28,000
this guy, we wouldn't do it this
way.

673
00:48:28,000 --> 00:48:31,000
We do symmetry,
and actually we'd rotate the

674
00:48:31,000 --> 00:48:34,000
thing so that our spherical cap
was actually centered on the z

675
00:48:34,000 --> 00:48:37,000
axis because that would be a
much easier way to set it up.

676
00:48:37,000 --> 00:48:39,000
But, depending on what function
we are integrating,

677
00:48:39,000 --> 00:48:42,000
we can't always do that.