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All right, so the past few
weeks,
we've been looking at double
integrals and the plane,
00:00:27.000 --> 00:00:31.000
line integrals in the plane,
and will we are going to do now
00:00:31.000 --> 00:00:34.000
from now on basically until the
end of the term,
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will be very similar stuff,
but in space.
00:00:36.000 --> 00:00:41.000
So, we are going to learn how
to do triple integrals in space,
00:00:41.000 --> 00:00:43.000
flux in space,
work in space,
00:00:43.000 --> 00:00:46.000
divergence, curl,
all that.
00:00:46.000 --> 00:00:49.000
So,
that means, basically,
00:00:49.000 --> 00:00:52.000
if you were really on top of
what we've been doing these past
00:00:52.000 --> 00:00:55.000
few weeks,
then it will be just the same
00:00:55.000 --> 00:00:58.000
with one more coordinate.
And, you will see there are
00:00:58.000 --> 00:01:00.000
some differences.
But, conceptually,
00:01:00.000 --> 00:01:04.000
it's pretty similar.
There are a few tricky things,
00:01:04.000 --> 00:01:06.000
though.
Now, that also means that if
00:01:06.000 --> 00:01:10.000
there is stuff that you are not
sure about in the plane,
00:01:10.000 --> 00:01:14.000
then I encourage you to review
the material that we've done
00:01:14.000 --> 00:01:18.000
over the past few weeks to make
sure that everything in the
00:01:18.000 --> 00:01:22.000
plane is completely clear to you
because it will be much harder
00:01:22.000 --> 00:01:26.000
to understand stuff in space if
things are still shaky in the
00:01:26.000 --> 00:01:30.000
plane.
OK, so the plan is we're going
00:01:30.000 --> 00:01:36.000
to basically go through the same
stuff, but in space.
00:01:36.000 --> 00:01:45.000
So, it shouldn't be surprising
that we will start today with
00:01:45.000 --> 00:01:52.000
triple integrals.
OK, so the way triple integrals
00:01:52.000 --> 00:01:58.000
work is if I give you a function
of three variables,
00:01:58.000 --> 00:02:02.000
x, y, z,
and I give you some region in
00:02:02.000 --> 00:02:07.000
space,
so, some solid,
00:02:07.000 --> 00:02:15.000
then I can take the integral
over this region over function f
00:02:15.000 --> 00:02:20.000
dV where dV stands for the
volume element.
00:02:20.000 --> 00:02:24.000
OK, so what it means is we will
just take every single little
00:02:24.000 --> 00:02:28.000
piece of our solid,
take the value of f there,
00:02:28.000 --> 00:02:30.000
multiply by the small volume of
each little piece,
00:02:30.000 --> 00:02:35.000
and sum all these things
together.
00:02:35.000 --> 00:02:40.000
And,
so this volume element here,
00:02:40.000 --> 00:02:44.000
well, for example,
if you are doing the integral
00:02:44.000 --> 00:02:48.000
in rectangular coordinates,
that will become dx dy dz or
00:02:48.000 --> 00:02:54.000
any permutation of that because,
of course, we have lots of
00:02:54.000 --> 00:02:59.000
possible orders of integration
to choose from.
00:02:59.000 --> 00:03:08.000
So, rather than bore you with
theory and all sorts of
00:03:08.000 --> 00:03:15.000
complicated things,
let's just do examples.
00:03:15.000 --> 00:03:18.000
And, you will see, basically,
if you understand how to set up
00:03:18.000 --> 00:03:21.000
iterated integrals into
variables,
00:03:21.000 --> 00:03:23.000
that you basically understand
how to do them in three
00:03:23.000 --> 00:03:27.000
variables.
You just have to be a bit more
00:03:27.000 --> 00:03:30.000
careful.
And, there's one more step.
00:03:30.000 --> 00:03:35.000
OK, so let's take our first
triple integral to be on the
00:03:35.000 --> 00:03:36.000
region.
So, of course,
00:03:36.000 --> 00:03:37.000
there's two different things as
always.
00:03:37.000 --> 00:03:39.000
There is the region of
integration and there's the
00:03:39.000 --> 00:03:42.000
function we are integrating.
Now, the function we are
00:03:42.000 --> 00:03:44.000
integrating, well,
it will come in handy when you
00:03:44.000 --> 00:03:46.000
actually try to evaluate the
integral.
00:03:46.000 --> 00:03:49.000
But, as you can see,
probably, the new part is
00:03:49.000 --> 00:03:52.000
really hard to set it up.
So, the function won't really
00:03:52.000 --> 00:03:55.000
matter that much for me.
So, in the examples I'll do
00:03:55.000 --> 00:03:58.000
today, functions will be kind of
silly.
00:03:58.000 --> 00:04:04.000
So, for example,
let's say that we want to look
00:04:04.000 --> 00:04:13.000
at the region between two
paraboloids, one given by z = x
00:04:13.000 --> 00:04:20.000
^2 y ^2.
The other is z = 4 - x ^2 - y
00:04:20.000 --> 00:04:22.000
^2.
And, so, I haven't given you,
00:04:22.000 --> 00:04:26.000
yet, the function to integrate.
OK, this is not the function to
00:04:26.000 --> 00:04:28.000
integrate.
This is what describes the
00:04:28.000 --> 00:04:32.000
region where I will integrate my
function.
00:04:32.000 --> 00:04:38.000
And, let's say that I just want
to find the volume of this
00:04:38.000 --> 00:04:43.000
region, which is the triple
integral of just one dV.
00:04:43.000 --> 00:04:46.000
OK, similarly,
remember, when we try to find
00:04:46.000 --> 00:04:49.000
the area of the region in the
plane, we are just integrating
00:04:49.000 --> 00:04:51.000
one dA.
Here we integrate one dV.
00:04:51.000 --> 00:04:55.000
that will give us the volume.
Now, I know that you can
00:04:55.000 --> 00:04:59.000
imagine how to actually do this
one as a double integral.
00:04:59.000 --> 00:05:02.000
But, the goal of the game is to
set up the triple integral.
00:05:02.000 --> 00:05:05.000
It's not actually to find the
volume.
00:05:05.000 --> 00:05:12.000
So, what does that look like?
Well, z = x ^2 y ^2,
00:05:12.000 --> 00:05:16.000
that's one of our favorite
paraboloids.
00:05:16.000 --> 00:05:22.000
That's something that looks
like a parabola with its bottom
00:05:22.000 --> 00:05:28.000
at the origin that you spin
about the z axis.
00:05:28.000 --> 00:05:32.000
And, z equals four minus x
squared minus y squared,
00:05:32.000 --> 00:05:36.000
well, that's also a paraboloid.
But, this one is pointing down,
00:05:36.000 --> 00:05:40.000
and when you take x equals y
equals zero, you get z equals
00:05:40.000 --> 00:05:44.000
four.
So, it starts at four,
00:05:44.000 --> 00:05:52.000
and it goes down like that.
OK, so the solid that we'd like
00:05:52.000 --> 00:05:56.000
to consider is what's in between
in here.
00:05:56.000 --> 00:06:00.000
So, it has a curvy top which is
this downward paraboloid,
00:06:00.000 --> 00:06:04.000
a curvy bottom which is the
other paraboloid.
00:06:04.000 --> 00:06:08.000
And, what about the sides?
Well, do you have any idea what
00:06:08.000 --> 00:06:11.000
we get here?
Yeah, it's going to be a circle
00:06:11.000 --> 00:06:15.000
because entire picture is
invariant by rotation about the
00:06:15.000 --> 00:06:17.000
z axis.
So, if you look at the picture
00:06:17.000 --> 00:06:20.000
just, say, in the yz plane,
you get this point and that
00:06:20.000 --> 00:06:24.000
point.
And, when you rotate everything
00:06:24.000 --> 00:06:30.000
around the z axis,
you will just get a circle
00:06:30.000 --> 00:06:33.000
here.
OK, so our goal is to find the
00:06:33.000 --> 00:06:36.000
volume of this thing,
and there's lots of things I
00:06:36.000 --> 00:06:38.000
could do to simplify the
calculation,
00:06:38.000 --> 00:06:41.000
or even not do it as a triple
integral at all.
00:06:41.000 --> 00:06:46.000
But, I want to actually set it
up as a triple integral just to
00:06:46.000 --> 00:06:50.000
show how we do that.
OK, so the first thing we need
00:06:50.000 --> 00:06:54.000
to do is choose an order of
integration.
00:06:54.000 --> 00:06:56.000
And, here, well,
I don't know if you can see it
00:06:56.000 --> 00:07:00.000
yet, but hopefully soon that
will be intuitive to you.
00:07:00.000 --> 00:07:04.000
I claim that I would like to
start by integrating first over
00:07:04.000 --> 00:07:05.000
z.
What's the reason for that?
00:07:05.000 --> 00:07:09.000
Well, the reason is if I give
you x and y, then you can find
00:07:09.000 --> 00:07:13.000
quickly, what's the bottom and
top values of z for that choice
00:07:13.000 --> 00:07:18.000
of x and y?
OK, so if I have x and y given,
00:07:18.000 --> 00:07:25.000
then I can find above that:
what is the bottom z and the
00:07:25.000 --> 00:07:33.000
top z corresponding to the
vertical line above that point?
00:07:33.000 --> 00:07:39.000
The portion of it that's inside
our solid, so somehow,
00:07:39.000 --> 00:07:45.000
there's a bottom z and a top z.
And, so the top z is actually
00:07:45.000 --> 00:07:49.000
on the downward paraboloid.
So, it's four minus x squared
00:07:49.000 --> 00:07:52.000
minus y squared.
The bottom value of z is x
00:07:52.000 --> 00:07:58.000
squared plus y squared.
OK, so if I want to start to
00:07:58.000 --> 00:08:04.000
set this up, I will write the
triple integral.
00:08:04.000 --> 00:08:09.000
And then, so let's say I'm
going to do it dz first,
00:08:09.000 --> 00:08:11.000
and then, say,
dy dx.
00:08:11.000 --> 00:08:16.000
It doesn't really matter.
So then, for a given value of x
00:08:16.000 --> 00:08:19.000
and y, I claim z goes from the
bottom surface.
00:08:19.000 --> 00:08:23.000
The bottom face is z equals x
squared plus y squared.
00:08:23.000 --> 00:08:29.000
The top face is four minus x
squared minus y squared.
00:08:29.000 --> 00:08:35.000
OK, is that OK with everyone?
Yeah?
00:08:35.000 --> 00:08:43.000
Any questions so far?
Yes?
00:08:43.000 --> 00:08:45.000
Why did I start with z?
That's a very good question.
00:08:45.000 --> 00:08:49.000
So, I can choose whatever order
I want, but let's say I did x
00:08:49.000 --> 00:08:50.000
first .
Then, to find the inner
00:08:50.000 --> 00:08:53.000
integral bounds,
I would need to say, OK,
00:08:53.000 --> 00:08:56.000
I've chosen values of,
see, in the inner integral,
00:08:56.000 --> 00:08:59.000
you've fixed the two other
variables,
00:08:59.000 --> 00:09:01.000
and you're just going to vary
that one.
00:09:01.000 --> 00:09:02.000
And, you need to find bounds
for it.
00:09:02.000 --> 00:09:05.000
So, if I integrate over x
first, I have to solve,
00:09:05.000 --> 00:09:10.000
answer the following question.
Say I'm given values of y and z.
00:09:10.000 --> 00:09:14.000
What are the bounds for x?
So, that would mean I'm slicing
00:09:14.000 --> 00:09:18.000
my solid by lines that are
parallel to the x axis.
00:09:18.000 --> 00:09:21.000
And, see, it's kind of hard to
find, what are the values of x
00:09:21.000 --> 00:09:24.000
at the front and at the back?
I mean, it's possible,
00:09:24.000 --> 00:09:27.000
but it's easier to actually
first look for z at the top and
00:09:27.000 --> 00:09:33.000
bottom.
Yes?
00:09:33.000 --> 00:09:36.000
dy dx, or dx dy?
No, it's completely at random.
00:09:36.000 --> 00:09:39.000
I mean, you can see x and y
play symmetric roles.
00:09:39.000 --> 00:09:43.000
So, if you look at it,
it's reasonably clear that z
00:09:43.000 --> 00:09:49.000
should be the easiest one to set
up first for what comes next.
00:09:49.000 --> 00:09:54.000
xy or yx, it's the same.
Yes?
00:09:54.000 --> 00:09:56.000
Yes, it will be easier to use
cylindrical coordinates.
00:09:56.000 --> 00:10:03.000
I'll get to that just as soon
as I'm done with this one.
00:10:03.000 --> 00:10:07.000
OK, so let's continue a bit
with that.
00:10:07.000 --> 00:10:11.000
And, as you mentioned,
actually we don't actually want
00:10:11.000 --> 00:10:14.000
to do it with xy in the end.
In a few minutes,
00:10:14.000 --> 00:10:16.000
we will actually switch to
cylindrical coordinates.
00:10:16.000 --> 00:10:18.000
But, for now,
we don't even know what they
00:10:18.000 --> 00:10:20.000
are.
OK, so I've done the inner
00:10:20.000 --> 00:10:25.000
integral by looking at,
you know, if I slice by
00:10:25.000 --> 00:10:28.000
vertical lines,
what is the top?
00:10:28.000 --> 00:10:31.000
What is the bottom for a given
value of x and y?
00:10:31.000 --> 00:10:36.000
So, the bounds in the inner
integral depend on both the
00:10:36.000 --> 00:10:41.000
middle and outer variables.
Next, I need to figure out what
00:10:41.000 --> 00:10:44.000
values of x and y I will be
interested in.
00:10:44.000 --> 00:10:47.000
And, the answer for that is,
well, the values of x and y
00:10:47.000 --> 00:10:51.000
that I want to look at are all
those that are in the shade of
00:10:51.000 --> 00:10:53.000
my region.
So, in fact,
00:10:53.000 --> 00:10:57.000
to set up the middle and outer
bounds, what I want to do is
00:10:57.000 --> 00:11:04.000
project my solid.
So, my solid looks like this
00:11:04.000 --> 00:11:09.000
kind of thing.
And, I don't really know how to
00:11:09.000 --> 00:11:13.000
call it.
But, what's interesting now is
00:11:13.000 --> 00:11:18.000
I want to look at the shadow
that it casts in the xy plane.
00:11:18.000 --> 00:11:22.000
OK, and, of course,
that shadow will just be the
00:11:22.000 --> 00:11:27.000
disk that's directly below this
disk here that's separating the
00:11:27.000 --> 00:11:34.000
two halves of the solid.
And so, now I will want to
00:11:34.000 --> 00:11:41.000
integrate over,
I want to look at all the xy's,
00:11:41.000 --> 00:11:46.000
x and y, in the shadow.
So, now I'm left with,
00:11:46.000 --> 00:11:48.000
actually, something we've
already done,
00:11:48.000 --> 00:11:52.000
namely setting up a double
integral over x and y.
00:11:52.000 --> 00:11:55.000
So, if it helps,
here, we don't strictly need
00:11:55.000 --> 00:11:59.000
it, but if it helps,
it could be useful to actually
00:11:59.000 --> 00:12:03.000
draw a picture of this shadow in
the xy plane.
00:12:03.000 --> 00:12:14.000
So, here it would just look,
again, like a disk,
00:12:14.000 --> 00:12:18.000
and set it up.
Now, the question is,
00:12:18.000 --> 00:12:22.000
how do we find the size of this
disk, the size of the shadow?
00:12:22.000 --> 00:12:28.000
Well, basically we have to
figure out where our two
00:12:28.000 --> 00:12:37.000
paraboloids intersect.
There's nothing else.
00:12:37.000 --> 00:12:49.000
OK, so, one way how to find the
shadow in the xy plane -- --
00:12:49.000 --> 00:12:53.000
well,
here we actually know the
00:12:53.000 --> 00:12:56.000
answer a priori,
but even if we didn't,
00:12:56.000 --> 00:12:59.000
we could just say,
well, our region lives wherever
00:12:59.000 --> 00:13:03.000
the bottom surface is below the
top surface,
00:13:03.000 --> 00:13:11.000
OK, so we want to look at
things wherever bottom value of
00:13:11.000 --> 00:13:15.000
z is less than the top value of
z,
00:13:15.000 --> 00:13:18.000
I mean, less or less than or
equal, that's the same thing.
00:13:18.000 --> 00:13:24.000
So, if the bottom value of z is
x squared plus y squared should
00:13:24.000 --> 00:13:28.000
be less than four minus x
squared minus y squared,
00:13:28.000 --> 00:13:33.000
and if you solve for that,
then you will get,
00:13:33.000 --> 00:13:34.000
well, so let's move these guys
over here.
00:13:34.000 --> 00:13:37.000
You'll get two x squared plus
two y squared less than four.
00:13:37.000 --> 00:13:42.000
That becomes x squared plus y
squared less than two.
00:13:42.000 --> 00:13:52.000
So, that means that's a disk of
radius square root of two,
00:13:52.000 --> 00:13:56.000
OK?
So, we kind of knew in advance
00:13:56.000 --> 00:14:01.000
it was going to be a disk,
but what we've learned now is
00:14:01.000 --> 00:14:05.000
that this radius is square root
of two.
00:14:05.000 --> 00:14:08.000
So, if we want to set up,
if we really want to set it up
00:14:08.000 --> 00:14:13.000
using dy dx like they started,
then we can do it because we
00:14:13.000 --> 00:14:16.000
know,
so, for the middle integral,
00:14:16.000 --> 00:14:19.000
now,
we want to fix a value of x.
00:14:19.000 --> 00:14:21.000
And, for that fixed value of x,
we want to figure out the
00:14:21.000 --> 00:14:25.000
bounds for y.
Well, the answer is y goes from
00:14:25.000 --> 00:14:26.000
here to here.
What's here?
00:14:26.000 --> 00:14:31.000
Well, here, y is square root of
two minus x squared.
00:14:31.000 --> 00:14:36.000
And, here it's negative square
root of two minus x squared.
00:14:36.000 --> 00:14:40.000
So, y will go from negative
square root of two minus x
00:14:40.000 --> 00:14:46.000
squared to positive square root.
And then, x will go from
00:14:46.000 --> 00:14:52.000
negative root two to root two.
OK, if that's not completely
00:14:52.000 --> 00:14:55.000
clear to you,
then I encourage you to go over
00:14:55.000 --> 00:14:58.000
how we set up double integrals
again.
00:14:58.000 --> 00:15:02.000
OK, does that make sense,
kind of?
00:15:02.000 --> 00:15:17.000
Yeah?
Well, so, when we set up,
00:15:17.000 --> 00:15:20.000
remember, we are setting up a
double integral,
00:15:20.000 --> 00:15:23.000
dy dx here.
So, when we do it dy dx,
00:15:23.000 --> 00:15:27.000
it means we slice this region
of a plane by vertical line
00:15:27.000 --> 00:15:30.000
segments.
So, this middle guy would be
00:15:30.000 --> 00:15:33.000
what used to be the inner
integral.
00:15:33.000 --> 00:15:36.000
So, in the inner,
remember, you fix the value of
00:15:36.000 --> 00:15:39.000
x, and you ask yourself,
what is the range of values of
00:15:39.000 --> 00:15:43.000
y in my region?
So, y goes from here to here,
00:15:43.000 --> 00:15:46.000
and what here and here are
depends on the value of x.
00:15:46.000 --> 00:15:48.000
How?
Well, we have to find the
00:15:48.000 --> 00:15:50.000
relation between x and y at
these points.
00:15:50.000 --> 00:15:53.000
These points are on the circle
of radius root two.
00:15:53.000 --> 00:15:56.000
So, if you want this circle
maybe I should have written,
00:15:56.000 --> 00:15:58.000
is x squared plus y squared
equals two.
00:15:58.000 --> 00:16:03.000
And, if you solve for y,
given x, you get plus minus
00:16:03.000 --> 00:16:07.000
root of two minus x squared,
OK?
00:16:07.000 --> 00:16:11.000
Yes?
Is there a way to compute this
00:16:11.000 --> 00:16:13.000
with symmetry?
Well, certainly,
00:16:13.000 --> 00:16:15.000
yeah, this solid looks
sufficiently symmetric,
00:16:15.000 --> 00:16:17.000
but actually you could
certainly,
00:16:17.000 --> 00:16:19.000
if you don't want to do the
whole disk,
00:16:19.000 --> 00:16:21.000
you could just do quarter
disks,
00:16:21.000 --> 00:16:25.000
and multiply by four.
You could even just look at the
00:16:25.000 --> 00:16:29.000
lower half of the solid,
and multiply them by two,
00:16:29.000 --> 00:16:33.000
so, total by eight.
So, yeah, certainly there's
00:16:33.000 --> 00:16:36.000
lots of ways to make it slightly
easier by using symmetry.
00:16:36.000 --> 00:16:39.000
Now, the most spectacular way
to use symmetry here,
00:16:39.000 --> 00:16:41.000
of course, is to use that we
have this rotation symmetry and
00:16:41.000 --> 00:16:45.000
switch,
actually, not do this guy in xy
00:16:45.000 --> 00:16:50.000
coordinates but instead in polar
coordinates.
00:16:50.000 --> 00:17:08.000
So -- So, the smarter thing to
do would be to use polar
00:17:08.000 --> 00:17:21.000
coordinates instead of x and y.
Of course, we want to keep z.
00:17:21.000 --> 00:17:23.000
I mean, we are very happy with
z the way it is.
00:17:23.000 --> 00:17:28.000
But, we'll just change x and y
to R cos theta,
00:17:28.000 --> 00:17:31.000
R sine theta,
OK, because,
00:17:31.000 --> 00:17:37.000
well, let's see actually how we
would evaluate this guy.
00:17:37.000 --> 00:17:46.000
So, well actually, let's not.
It's kind of boring.
00:17:46.000 --> 00:17:50.000
So, let me just point out one
small thing here,
00:17:50.000 --> 00:17:54.000
sorry, before I do that.
So, if you start computing the
00:17:54.000 --> 00:17:59.000
inner integral,
OK, so let me not do that yet,
00:17:59.000 --> 00:18:03.000
sorry,
so if you try to compute the
00:18:03.000 --> 00:18:06.000
inner integral,
you'll be integrating from x
00:18:06.000 --> 00:18:11.000
squared plus y squared to four
minus x squared minus y squared
00:18:11.000 --> 00:18:15.000
dz.
Well, that will integrate to z
00:18:15.000 --> 00:18:22.000
between these two bounds.
So, you will get four minus two
00:18:22.000 --> 00:18:27.000
x squared minus two y squared.
Now, when you put that into the
00:18:27.000 --> 00:18:33.000
remaining ones,
you'll get something that's
00:18:33.000 --> 00:18:41.000
probably not very pleasant of
four minus two x squared minus
00:18:41.000 --> 00:18:49.000
two y squared dy dx.
And here, you see that to
00:18:49.000 --> 00:18:56.000
evaluate this,
you would switch to polar
00:18:56.000 --> 00:18:59.000
coordinates.
Oh, by the way,
00:18:59.000 --> 00:19:04.000
so if your initial instincts
had been to,
00:19:04.000 --> 00:19:06.000
given that you just want the
volume,
00:19:06.000 --> 00:19:09.000
you could also have found the
volume just by doing a double
00:19:09.000 --> 00:19:12.000
integral of the height between
the top and bottom.
00:19:12.000 --> 00:19:14.000
Well, you would just have
gotten this, right,
00:19:14.000 --> 00:19:17.000
because this is the height
between top and bottom.
00:19:17.000 --> 00:19:21.000
So, it's all the same.
It doesn't really matter.
00:19:21.000 --> 00:19:23.000
But with this,
of course, we will be able to
00:19:23.000 --> 00:19:26.000
integrate all sorts of
functions, not just one over the
00:19:26.000 --> 00:19:31.000
solid.
So, we will be able to do much
00:19:31.000 --> 00:19:35.000
more than just volumes.
OK, so let's see,
00:19:35.000 --> 00:19:37.000
how do we do it with polar
coordinates instead?
00:19:37.000 --> 00:19:53.000
Well, so -- Well,
that would become,
00:19:53.000 --> 00:20:02.000
so let's see.
So, I want to keep dz.
00:20:02.000 --> 00:20:10.000
But then, dx dy or dy dx would
become r dr d theta.
00:20:10.000 --> 00:20:13.000
And, if I try to set up the
bounds, well,
00:20:13.000 --> 00:20:17.000
I probably shouldn't keep this
x squared plus y squared around.
00:20:17.000 --> 00:20:20.000
But, x squared plus y squared
is easy in terms of r and theta.
00:20:20.000 --> 00:20:26.000
That's just r squared.
OK, I mean, in general I could
00:20:26.000 --> 00:20:28.000
have something that depends also
on theta.
00:20:28.000 --> 00:20:32.000
That's perfectly legitimate.
But here, it simplifies,
00:20:32.000 --> 00:20:36.000
and this guy up here,
four minus x squared minus y
00:20:36.000 --> 00:20:40.000
squared becomes four minus r
squared.
00:20:40.000 --> 00:20:43.000
And now, the integral that we
have to do over r and theta,
00:20:43.000 --> 00:20:45.000
well, we look again at the
shadow.
00:20:45.000 --> 00:20:48.000
The shadow is still a disk of
radius root two.
00:20:48.000 --> 00:20:52.000
That hasn't changed.
And now, we know how to set up
00:20:52.000 --> 00:20:54.000
this integral in polar
coordinates.
00:20:54.000 --> 00:21:01.000
r goes from zero to root two,
and theta goes from zero to two
00:21:01.000 --> 00:21:11.000
pi.
OK, and now it becomes actually
00:21:11.000 --> 00:21:20.000
easier to evaluate.
OK, so now we have actually a
00:21:20.000 --> 00:21:24.000
name for this because we're
doing it in space.
00:21:24.000 --> 00:21:28.000
So, these are called,
actually, cylindrical
00:21:28.000 --> 00:21:30.000
coordinates.
So, in fact,
00:21:30.000 --> 00:21:35.000
you already knew about
cylindrical coordinates even if
00:21:35.000 --> 00:21:40.000
you did not know the name.
OK, so the idea of cylindrical
00:21:40.000 --> 00:21:45.000
coordinates is that instead of
x, y, and z, to locate a point
00:21:45.000 --> 00:21:48.000
in space, you will use three
coordinates.
00:21:48.000 --> 00:22:00.000
One of them is basically how
high it is above the xy plane.
00:22:00.000 --> 00:22:04.000
So, that will be z.
And then, you will use polar
00:22:04.000 --> 00:22:08.000
coordinates for the projection
of your point on the xy plane.
00:22:08.000 --> 00:22:12.000
So, r will be the distance from
the z axis.
00:22:12.000 --> 00:22:17.000
And theta will be the angle
from the x axis
00:22:17.000 --> 00:22:21.000
counterclockwise.
So, the one thing to be careful
00:22:21.000 --> 00:22:24.000
about is because of the usual
convention, that we make the x
00:22:24.000 --> 00:22:27.000
axis point toward us.
Theta equals zero is no longer
00:22:27.000 --> 00:22:30.000
to the right.
Now, theta equals zero is to
00:22:30.000 --> 00:22:34.000
the front, and the angel is
measured from the front
00:22:34.000 --> 00:22:39.000
counterclockwise.
OK, so,
00:22:39.000 --> 00:22:41.000
and of course,
if you want to know how to
00:22:41.000 --> 00:22:44.000
convert between x,
y, z and r theta z,
00:22:44.000 --> 00:22:49.000
well, the formulas are just the
same as in usual polar
00:22:49.000 --> 00:22:52.000
coordinates.
R cos theta,
00:22:52.000 --> 00:22:56.000
r sine theta,
and z remain z.
00:22:56.000 --> 00:22:59.000
OK, so why are these called
cylindrical coordinates,
00:22:59.000 --> 00:23:02.000
by the way?
Well, let's say that I gave you
00:23:02.000 --> 00:23:07.000
the equation r equals a,
where a is some constant.
00:23:07.000 --> 00:23:12.000
Say r equals one, for example.
So, r equals one in 2D,
00:23:12.000 --> 00:23:15.000
that used to be just a circle
of radius one.
00:23:15.000 --> 00:23:19.000
Now, in space,
a single equation actually
00:23:19.000 --> 00:23:23.000
defines a surface,
not just a curve anymore.
00:23:23.000 --> 00:23:26.000
And, the set of points where r
is a, well, that's all the
00:23:26.000 --> 00:23:29.000
points that are distance a from
the z axis.
00:23:29.000 --> 00:23:34.000
So, in fact,
what you get this way is a
00:23:34.000 --> 00:23:41.000
cylinder of radius a centered on
the z axis.
00:23:41.000 --> 00:23:48.000
OK, so that's why they are
called cylindrical coordinates.
00:23:48.000 --> 00:23:51.000
By the way, so now,
similarly, if you look at the
00:23:51.000 --> 00:23:55.000
equation theta equals some given
value, well, so that used to be
00:23:55.000 --> 00:23:59.000
just a ray from the origin.
Now, that becomes a vertical
00:23:59.000 --> 00:24:01.000
half plane.
For example,
00:24:01.000 --> 00:24:04.000
if I set the value of theta and
let r and z vary,
00:24:04.000 --> 00:24:08.000
well, r is always positive,
but basically that means I am
00:24:08.000 --> 00:24:13.000
taking a vertical plane that
comes out in this direction.
00:24:13.000 --> 00:24:18.000
OK, any questions about
cylindrical coordinates?
00:24:18.000 --> 00:24:27.000
Yes?
Yeah, so I'm saying when you
00:24:27.000 --> 00:24:30.000
fix theta, you get only a half
plane, not a full plane.
00:24:30.000 --> 00:24:33.000
I mean, it goes all the way up
and down, but it doesn't go back
00:24:33.000 --> 00:24:35.000
to the other side of the z axis.
Why?
00:24:35.000 --> 00:24:39.000
That's because r is always
positive by convention.
00:24:39.000 --> 00:24:41.000
So, for example,
here, we say theta is zero.
00:24:41.000 --> 00:24:44.000
At the back, we say theta is pi.
We don't say theta is zero and
00:24:44.000 --> 00:24:47.000
r is negative.
We say r is positive and theta
00:24:47.000 --> 00:24:50.000
is pi.
It's a convention, largely.
00:24:50.000 --> 00:24:52.000
But, sticking with this
convention really will help you
00:24:52.000 --> 00:24:54.000
to set up the integrals
properly.
00:24:54.000 --> 00:24:58.000
I mean, otherwise there is just
too much risk for mistakes.
00:24:58.000 --> 00:25:08.000
Yes?
Well, so the question is if I
00:25:08.000 --> 00:25:11.000
were to use symmetry to do this
one, would I multiply by four or
00:25:11.000 --> 00:25:13.000
by two?
Well, it depends on how much
00:25:13.000 --> 00:25:16.000
symmetry you are using.
So, I mean, it's your choice.
00:25:16.000 --> 00:25:19.000
You can multiply by two,
by four, by eight depending on
00:25:19.000 --> 00:25:22.000
how much you cut it.
So, it depends on what symmetry
00:25:22.000 --> 00:25:24.000
you use, if you use symmetry
between top and bottom you'd
00:25:24.000 --> 00:25:27.000
say, well, the volume is twice
the lower half.
00:25:27.000 --> 00:25:30.000
If you use the left and right
half, you would say it's twice
00:25:30.000 --> 00:25:34.000
each half.
If you cut it into four pieces,
00:25:34.000 --> 00:25:37.000
and so on.
So, and again,
00:25:37.000 --> 00:25:41.000
you don't have to use the
symmetry.
00:25:41.000 --> 00:25:44.000
If you don't think of using
polar coordinates,
00:25:44.000 --> 00:25:46.000
then it can save you from
doing,
00:25:46.000 --> 00:25:47.000
you know, you can just start at
zero here and here,
00:25:47.000 --> 00:26:01.000
and simplify things a tiny bit.
But, OK, yes?
00:26:01.000 --> 00:26:04.000
So, to define a vertical full
plane, well, first of all it
00:26:04.000 --> 00:26:07.000
depends on whether it passes
through the z axis or not.
00:26:07.000 --> 00:26:09.000
If it doesn't,
then you'd have to remember how
00:26:09.000 --> 00:26:13.000
you do in polar coordinates.
I mean, basically the answer
00:26:13.000 --> 00:26:16.000
is, if you have a vertical
plane, so, it doesn't depend on
00:26:16.000 --> 00:26:18.000
z.
The equation does not involve z.
00:26:18.000 --> 00:26:21.000
It only involves r and theta.
And, how it involves r and
00:26:21.000 --> 00:26:24.000
theta is exactly the same as
when you do a line in polar
00:26:24.000 --> 00:26:27.000
coordinates in the plane.
So, if it's a line passing
00:26:27.000 --> 00:26:29.000
through the origin,
you say, well,
00:26:29.000 --> 00:26:32.000
theta is either some value or
the other one.
00:26:32.000 --> 00:26:33.000
If it's a line that doesn't
passes to the origin,
00:26:33.000 --> 00:26:38.000
but it's more tricky.
But hopefully you've seen how
00:26:38.000 --> 00:26:49.000
to do that.
OK, let's move on a bit.
00:26:49.000 --> 00:26:53.000
So, one thing to know,
I mean, basically,
00:26:53.000 --> 00:26:57.000
the important thing to remember
is that the volume element in
00:26:57.000 --> 00:27:05.000
cylindrical coordinates,
well, dx dy dz becomes r dr d
00:27:05.000 --> 00:27:08.000
theta dz.
And, that shouldn't be
00:27:08.000 --> 00:27:12.000
surprising because that's just
dx dy becomes r dr d theta.
00:27:12.000 --> 00:27:18.000
And, dz remains dz.
I mean, so, the way to think
00:27:18.000 --> 00:27:19.000
about it,
if you want,
00:27:19.000 --> 00:27:25.000
is that if you take a little
piece of solid in space,
00:27:25.000 --> 00:27:31.000
so it has some height, delta z,
and it has a base which has
00:27:31.000 --> 00:27:36.000
some area delta A,
then the small volume, delta v,
00:27:36.000 --> 00:27:41.000
is equal to the area of a base
times the height.
00:27:41.000 --> 00:27:43.000
So, now, when you make the
things infinitely small,
00:27:43.000 --> 00:27:51.000
you will get dV is dA times dz,
and you can use whichever
00:27:51.000 --> 00:27:56.000
formula you want for area in the
xy plane.
00:27:56.000 --> 00:28:00.000
OK, now in practice,
you choose which order you
00:28:00.000 --> 00:28:03.000
integrate in.
As you have probably seen,
00:28:03.000 --> 00:28:07.000
a favorite of mine is z first
because very often you'll know
00:28:07.000 --> 00:28:10.000
what the top and bottom of your
solid look like,
00:28:10.000 --> 00:28:13.000
and then you will reduce to
just something in the xy plane.
00:28:13.000 --> 00:28:18.000
But, there might be situations
where it's actually easier to
00:28:18.000 --> 00:28:22.000
start first with dx dy or r dr d
theta, and then save dz for
00:28:22.000 --> 00:28:24.000
last.
I mean, if you seen how to,
00:28:24.000 --> 00:28:27.000
in single variable calculus,
the disk and shell methods for
00:28:27.000 --> 00:28:30.000
finding volumes,
that's exactly the dilemma of
00:28:30.000 --> 00:28:39.000
shells versus disks.
One of them is you do z first.
00:28:39.000 --> 00:28:49.000
The other is you do z last.
OK, so what are things we can
00:28:49.000 --> 00:28:56.000
do now with triple integrals?
Well, we can find the volume of
00:28:56.000 --> 00:29:00.000
solids by just integrating dV.
And, we've seen that.
00:29:00.000 --> 00:29:06.000
We can find the mass of a solid.
OK, so if we have a density,
00:29:06.000 --> 00:29:10.000
delta, which,
remember, delta is basically
00:29:10.000 --> 00:29:17.000
the mass divided by the volume.
OK, so the small mass element,
00:29:17.000 --> 00:29:23.000
maybe I should have written
that as dm, the mass element,
00:29:23.000 --> 00:29:27.000
is density times dV.
So now, this is the real
00:29:27.000 --> 00:29:29.000
physical density.
If you are given a material,
00:29:29.000 --> 00:29:33.000
usually, the density will be in
grams per cubic meter or cubic
00:29:33.000 --> 00:29:35.000
inch, or whatever.
I mean, there is tons of
00:29:35.000 --> 00:29:37.000
different units.
But, so then,
00:29:37.000 --> 00:29:43.000
the mass of your solid will be
just the triple integral of
00:29:43.000 --> 00:29:50.000
density, dV because you just sum
the mass of each little piece.
00:29:50.000 --> 00:29:53.000
And, of course,
if the density is one,
00:29:53.000 --> 00:29:55.000
then it just becomes the
volume.
00:29:55.000 --> 00:29:59.000
OK,
now, it shouldn't be surprising
00:29:59.000 --> 00:30:02.000
to you that we can also do
classics that we had seen in the
00:30:02.000 --> 00:30:05.000
plane such as the average value
of a function,
00:30:05.000 --> 00:30:07.000
the center of mass,
and moment of inertia.
00:30:38.000 --> 00:30:47.000
OK, so the average value of the
function f of x,
00:30:47.000 --> 00:30:51.000
y, z in the region,
r,
00:30:51.000 --> 00:30:57.000
that would be f bar,
would be one over the volume of
00:30:57.000 --> 00:31:02.000
the region times the triple
integral of f dV.
00:31:02.000 --> 00:31:09.000
Or, if we have a density,
and we want to take a weighted
00:31:09.000 --> 00:31:21.000
average -- Then we take one over
the mass where the mass is the
00:31:21.000 --> 00:31:32.000
triple integral of the density
times the triple integral of f
00:31:32.000 --> 00:31:36.000
density dV.
So, as particular cases,
00:31:36.000 --> 00:31:39.000
there is, again,
the notion of center of mass of
00:31:39.000 --> 00:31:41.000
the solid.
So, that's the point that
00:31:41.000 --> 00:31:44.000
somehow right in the middle of
the solid.
00:31:44.000 --> 00:31:48.000
That's the point mass by which
there is a point at which you
00:31:48.000 --> 00:31:53.000
should put point mass so that it
would be equivalent from the
00:31:53.000 --> 00:31:57.000
point of view of dealing with
forces and translation effects,
00:31:57.000 --> 00:32:03.000
of course, not for rotation.
But, so the center of mass of a
00:32:03.000 --> 00:32:09.000
solid is just given by taking
the average values of x,
00:32:09.000 --> 00:32:14.000
y, and z.
OK, so there is a special case
00:32:14.000 --> 00:32:21.000
where, so, x bar is one over the
mass times triple integral of x
00:32:21.000 --> 00:32:31.000
density dV.
And, same thing with y and z.
00:32:31.000 --> 00:32:33.000
And, of course,
very often, you can use
00:32:33.000 --> 00:32:37.000
symmetry to not have to compute
all three of them.
00:32:37.000 --> 00:32:39.000
For example,
if you look at this solid that
00:32:39.000 --> 00:32:41.000
we had, well,
I guess I've erased it now.
00:32:41.000 --> 00:32:43.000
But, if you remember what it
looked, well,
00:32:43.000 --> 00:32:45.000
it was pretty obvious that the
center of mass would be in the z
00:32:45.000 --> 00:32:47.000
axis.
So, no need to waste time
00:32:47.000 --> 00:32:49.000
considering x bar and y bar.
00:33:03.000 --> 00:33:08.000
And, in fact,
you can also find z bar by
00:33:08.000 --> 00:33:16.000
symmetry between the top and
bottom, and let you figure that
00:33:16.000 --> 00:33:18.000
out.
Of course, symmetry only works,
00:33:18.000 --> 00:33:20.000
I should say,
symmetry only works if the
00:33:20.000 --> 00:33:25.000
density is also symmetric.
If I had taken my guy to be
00:33:25.000 --> 00:33:31.000
heavier at the front than at the
back, then it would no longer be
00:33:31.000 --> 00:33:37.000
true that x bar would be zero.
OK, next on the list is moment
00:33:37.000 --> 00:33:40.000
of inertia.
Actually, in a way,
00:33:40.000 --> 00:33:45.000
moment of inertia in 3D is
easier conceptually than in 2D.
00:33:45.000 --> 00:33:49.000
So, why is that?
Well, because now the various
00:33:49.000 --> 00:33:53.000
flavors that we had come
together in a nice way.
00:33:53.000 --> 00:33:56.000
So, the moment of inertia of an
axis,
00:33:56.000 --> 00:33:58.000
sorry, with respect to an axis
would be,
00:33:58.000 --> 00:34:06.000
again, given by the triple
integral of the distance to the
00:34:06.000 --> 00:34:11.000
axis squared times density,
times dV.
00:34:11.000 --> 00:34:15.000
And, in particular,
we have our solid.
00:34:15.000 --> 00:34:19.000
And, we might skewer it using
any of the coordinate axes and
00:34:19.000 --> 00:34:22.000
then try to rotate it about one
of the axes.
00:34:22.000 --> 00:34:24.000
So, we have three different
possibilities,
00:34:24.000 --> 00:34:27.000
of course, the x,
y, or z axis.
00:34:27.000 --> 00:34:29.000
And, so now,
rotating about the z axis
00:34:29.000 --> 00:34:34.000
actually corresponds to when we
were just doing things for flat
00:34:34.000 --> 00:34:38.000
objects in the xy plane.
That corresponded to rotating
00:34:38.000 --> 00:34:40.000
about the origin.
So, secretly,
00:34:40.000 --> 00:34:42.000
we were saying we were rotating
about the point.
00:34:42.000 --> 00:34:44.000
But actually,
it was just rotating about the
00:34:44.000 --> 00:34:47.000
z axis.
Just I didn't want to introduce
00:34:47.000 --> 00:34:51.000
the z coordinate that we didn't
actually need at the time.
00:35:18.000 --> 00:35:23.000
So -- [APPLAUSE]
OK, so moment of inertia about
00:35:23.000 --> 00:35:28.000
the z axis, so,
what's the distance to the z
00:35:28.000 --> 00:35:31.000
axis?
Well, we've said that's exactly
00:35:31.000 --> 00:35:34.000
r.
That's the cylindrical
00:35:34.000 --> 00:35:39.000
coordinate, r.
So, the square of a distance is
00:35:39.000 --> 00:35:44.000
just r squared.
Now, if you didn't want to do
00:35:44.000 --> 00:35:49.000
it in cylindrical coordinates
then, of course,
00:35:49.000 --> 00:35:55.000
r squared is just x squared
plus y squared.
00:35:55.000 --> 00:35:58.000
Square of distance from the z
axis is just x squared plus y
00:35:58.000 --> 00:36:00.000
squared.
Similarly, now,
00:36:00.000 --> 00:36:04.000
if you want the distance from
the x axis, well,
00:36:04.000 --> 00:36:07.000
that will be y squared plus z
squared.
00:36:07.000 --> 00:36:09.000
OK, try to convince yourselves
of the picture,
00:36:09.000 --> 00:36:13.000
or else just argue by symmetry:
you know, if you change the
00:36:13.000 --> 00:36:18.000
positions of the axis.
So, moment of inertia about the
00:36:18.000 --> 00:36:25.000
x axis is the double integral of
y squared plus z squared delta
00:36:25.000 --> 00:36:29.000
dV.
And moment of inertia about the
00:36:29.000 --> 00:36:34.000
y axis is the same thing,
but now with x squared plus z
00:36:34.000 --> 00:36:36.000
squared.
And so, now,
00:36:36.000 --> 00:36:39.000
if you try to apply these
things for flat solids that are
00:36:39.000 --> 00:36:42.000
in the xy plane,
so where there's no z to look
00:36:42.000 --> 00:36:45.000
at,
well, you see these formulas
00:36:45.000 --> 00:36:48.000
become the old formulas that we
had.
00:36:48.000 --> 00:36:56.000
But now, they all fit together
in a more symmetric way.
00:36:56.000 --> 00:37:04.000
OK, any questions about that?
No?
00:37:04.000 --> 00:37:08.000
OK, so these are just formulas
to remember.
00:37:08.000 --> 00:37:21.000
So, OK, let's do an example.
Was there a question that I
00:37:21.000 --> 00:37:24.000
missed?
No?
00:37:24.000 --> 00:37:34.000
OK, so let's find the moment of
inertia about the z axis of a
00:37:34.000 --> 00:37:44.000
solid cone -- -- between z
equals a times r and z equals b.
00:37:44.000 --> 00:37:47.000
So, just to convince you that
it's a cone, so,
00:37:47.000 --> 00:37:52.000
z equals a times r means the
height is proportional to the
00:37:52.000 --> 00:37:57.000
distance from the z axis.
So, let's look at what we get
00:37:57.000 --> 00:38:00.000
if we just do it in the plane of
a blackboard.
00:38:00.000 --> 00:38:04.000
So, if I go to the right here,
r is just the distance from the
00:38:04.000 --> 00:38:06.000
x axis.
The height should be
00:38:06.000 --> 00:38:09.000
proportional with
proportionality factor A.
00:38:09.000 --> 00:38:14.000
So, that means I take a line
with slope A.
00:38:14.000 --> 00:38:16.000
If I'm on the left,
well, it's the same story
00:38:16.000 --> 00:38:19.000
except distance to the z axis is
still positive.
00:38:19.000 --> 00:38:22.000
So, I get the symmetric thing.
And, in fact,
00:38:22.000 --> 00:38:25.000
it doesn't matter which
vertical plane I do it in.
00:38:25.000 --> 00:38:28.000
This is the same if I rotate
about.
00:38:28.000 --> 00:38:32.000
See, there's no theta in here.
So, it's the same in all
00:38:32.000 --> 00:38:36.000
directions.
So, I claim it's a cone where
00:38:36.000 --> 00:38:43.000
the slope of the rays is A.
OK, and z equals b.
00:38:43.000 --> 00:38:50.000
Well, that just means we stop
in our horizontal plane at
00:38:50.000 --> 00:38:53.000
height b.
OK, so that's solid cone really
00:38:53.000 --> 00:38:58.000
just looks like this.
That's our solid.
00:38:58.000 --> 00:39:02.000
OK, so it has a flat top,
that circular top,
00:39:02.000 --> 00:39:08.000
and then the point is at v.
The tip of it is at the origin.
00:39:08.000 --> 00:39:12.000
So, let's try to compute its
moment of inertia about the z
00:39:12.000 --> 00:39:14.000
axis.
So, that means maybe this is
00:39:14.000 --> 00:39:16.000
like the top that you are going
to spin.
00:39:16.000 --> 00:39:21.000
And, it tells you how hard it
is to actually spin that top.
00:39:21.000 --> 00:39:24.000
Actually, that's also useful if
you're going to do mechanical
00:39:24.000 --> 00:39:27.000
engineering because if you are
trying to design gears,
00:39:27.000 --> 00:39:28.000
and things like that that will
rotate,
00:39:28.000 --> 00:39:31.000
you might want to know exactly
how much effort you'll have to
00:39:31.000 --> 00:39:33.000
put to actually get them to
spin,
00:39:33.000 --> 00:39:37.000
and whether you're actually
going to have a strong enough
00:39:37.000 --> 00:39:39.000
engine, or whatever,
to do it.
00:39:39.000 --> 00:39:41.000
OK, so what's the moment of
inertia of this guy?
00:39:41.000 --> 00:39:44.000
Well, that's the triple
integral of, well,
00:39:44.000 --> 00:39:49.000
we have to choose x squared
plus y squared or r squared.
00:39:49.000 --> 00:39:52.000
Let's see, I think I want to
use cylindrical coordinates to
00:39:52.000 --> 00:39:57.000
do that, given the shape.
So, we use r squared.
00:39:57.000 --> 00:40:02.000
I might have a density that
let's say the density is one.
00:40:02.000 --> 00:40:06.000
So, I don't have density.
I still have dV.
00:40:06.000 --> 00:40:13.000
Now, it will be my choice to
choose between doing the dz
00:40:13.000 --> 00:40:17.000
first or doing r dr d theta
first.
00:40:17.000 --> 00:40:20.000
Just to show you how it goes
the other way around,
00:40:20.000 --> 00:40:23.000
let me do it r dr d theta dz
this time.
00:40:23.000 --> 00:40:29.000
Then you can decide on a
case-by-case basis which one you
00:40:29.000 --> 00:40:33.000
like best.
OK, so if we do it in this
00:40:33.000 --> 00:40:36.000
direction, it means that in the
inner and middle integrals,
00:40:36.000 --> 00:40:40.000
we've fixed a value of z.
And, for that particular value
00:40:40.000 --> 00:40:45.000
of z, we'll be actually slicing
our solid by a horizontal plane,
00:40:45.000 --> 00:40:47.000
and looking at what we get,
OK?
00:40:47.000 --> 00:40:54.000
So, what does that look like?
Well, I fixed a value of z,
00:40:54.000 --> 00:40:59.000
and I slice my solid by a
horizontal plane.
00:40:59.000 --> 00:41:04.000
Well, I'm going to get a circle
certainly.
00:41:04.000 --> 00:41:07.000
What's the radius,
well, a disk actually,
00:41:07.000 --> 00:41:11.000
what's the radius of the disk?
Yeah, the radius of the disk
00:41:11.000 --> 00:41:14.000
should be z over a because the
equation of that cone,
00:41:14.000 --> 00:41:19.000
we said it's z equals ar.
So, if you flip it around,
00:41:19.000 --> 00:41:24.000
so, maybe I should switch to
another blackboard.
00:41:24.000 --> 00:41:33.000
So, the equation of a cone is z
equals ar, or equivalently r
00:41:33.000 --> 00:41:40.000
equals z over a.
So, for a given value of z,
00:41:40.000 --> 00:41:49.000
I will get, this guy will be a
disk of radius z over a.
00:41:49.000 --> 00:41:55.000
OK, so, moment of inertia is
going to be, well,
00:41:55.000 --> 00:41:59.000
we said r squared,
r dr d theta dz.
00:41:59.000 --> 00:42:02.000
Now, so, to set up the inner
and middle integrals,
00:42:02.000 --> 00:42:06.000
I just set up a double integral
over this disk of radius z over
00:42:06.000 --> 00:42:07.000
a.
So, it's easy.
00:42:07.000 --> 00:42:14.000
r goes from zero to z over a.
Theta goes from zero to 2pi.
00:42:14.000 --> 00:42:17.000
OK, and then,
well, if I set up the bounds
00:42:17.000 --> 00:42:20.000
for z, now it's my outer
variable.
00:42:20.000 --> 00:42:24.000
So, the question I have to ask
is what is the first slice?
00:42:24.000 --> 00:42:28.000
What is the last slice?
So, the bottommost value of z
00:42:28.000 --> 00:42:33.000
would be zero,
and the topmost would be b.
00:42:33.000 --> 00:42:37.000
And so, that's it I get.
So, exercise,
00:42:37.000 --> 00:42:42.000
it's not very hard.
Try to set it up the other way
00:42:42.000 --> 00:42:47.000
around with dz first and then r
dr d theta.
00:42:47.000 --> 00:42:49.000
It's pretty much the same level
of difficulty.
00:42:49.000 --> 00:42:52.000
I'm sure you can do both of
them.
00:42:52.000 --> 00:42:57.000
So, and also,
if you want to practice
00:42:57.000 --> 00:43:03.000
calculations,
you should end up getting pi b
00:43:03.000 --> 00:43:10.000
to the five over 10a to the four
if I got it right.
00:43:10.000 --> 00:43:14.000
OK, let me finish with one more
example.
00:43:14.000 --> 00:43:17.000
I'm trying to give you plenty
of practice because in case you
00:43:17.000 --> 00:43:19.000
haven't noticed,
Monday is a holiday.
00:43:19.000 --> 00:43:22.000
So, you don't have recitation
on Monday, which is good.
00:43:22.000 --> 00:43:25.000
But it means that there will be
lots of stuff to cover on
00:43:25.000 --> 00:43:47.000
Wednesday.
So -- Thank you.
00:43:47.000 --> 00:43:58.000
OK, so third example,
let's say that I want to just
00:43:58.000 --> 00:44:09.000
set up a triple integral for the
region where z is bigger than
00:44:09.000 --> 00:44:18.000
one minus y inside the unit ball
centered at the origin.
00:44:18.000 --> 00:44:24.000
So, the unit ball is just,
you know, well,
00:44:24.000 --> 00:44:30.000
stay inside of the unit sphere.
So, its equation,
00:44:30.000 --> 00:44:33.000
if you want,
would be x squared plus y
00:44:33.000 --> 00:44:35.000
squared plus z squared less than
one.
00:44:35.000 --> 00:44:37.000
OK, so that's one thing you
should remember.
00:44:37.000 --> 00:44:40.000
The equation of a sphere
centered at the origin is x
00:44:40.000 --> 00:44:44.000
squared plus y squared plus z
squared equals radius squared.
00:44:44.000 --> 00:44:48.000
And now, we are going to take
this plane, z equals one minus
00:44:48.000 --> 00:44:50.000
y.
So, if you think about it,
00:44:50.000 --> 00:44:52.000
it's parallel to the x axis
because there's no x in its
00:44:52.000 --> 00:44:55.000
coordinate in its equation.
At the origin,
00:44:55.000 --> 00:44:59.000
the height is one.
So, it starts right here at one.
00:44:59.000 --> 00:45:04.000
And, it slopes down with y with
slope one.
00:45:04.000 --> 00:45:07.000
OK, so it's a plane that comes
straight out here,
00:45:07.000 --> 00:45:10.000
and it intersects the sphere,
so here and here,
00:45:10.000 --> 00:45:13.000
but also at other points in
between.
00:45:13.000 --> 00:45:18.000
Any idea what kind of shape
this is?
00:45:18.000 --> 00:45:20.000
Well, it's an ellipse,
but it's even more than that.
00:45:20.000 --> 00:45:23.000
It's also a circle.
If you slice a sphere by a
00:45:23.000 --> 00:45:25.000
plane, you always get a circle.
But, of course,
00:45:25.000 --> 00:45:28.000
it's a slanted circle.
So, if you look at it in the xy
00:45:28.000 --> 00:45:31.000
plane, if you project it to the
xy plane, that you will get an
00:45:31.000 --> 00:45:35.000
ellipse.
OK, so we want to look at this
00:45:35.000 --> 00:45:38.000
guy in here.
So, how do we do that?
00:45:38.000 --> 00:45:42.000
Well, so maybe I should
actually draw quickly a picture.
00:45:42.000 --> 00:45:47.000
So, in the yz plane,
it looks just like this,
00:45:47.000 --> 00:45:51.000
OK?
But, if I look at it from above
00:45:51.000 --> 00:45:54.000
in the xy plane,
then its shadow,
00:45:54.000 --> 00:45:59.000
well, see, it will sit entirely
where y is positive.
00:45:59.000 --> 00:46:02.000
So, it sits entirely above
here, and it goes through here
00:46:02.000 --> 00:46:04.000
and here.
And, in fact,
00:46:04.000 --> 00:46:08.000
when you project that slanted
circle, now you will get an
00:46:08.000 --> 00:46:12.000
ellipse.
And, well, I don't really know
00:46:12.000 --> 00:46:20.000
how to draw it well,
but it should be something like
00:46:20.000 --> 00:46:24.000
this.
OK, so now if you want to try
00:46:24.000 --> 00:46:29.000
to set up that double integral,
sorry, the triple integral,
00:46:29.000 --> 00:46:37.000
well, so let's say we do it in
rectangular coordinates because
00:46:37.000 --> 00:46:41.000
we are really evil.
[LAUGHTER]
00:46:41.000 --> 00:46:43.000
So then, the bottom surface,
OK, so we do it with z first.
00:46:43.000 --> 00:46:46.000
So, the bottom surface is the
slanted plane.
00:46:46.000 --> 00:46:51.000
So, the bottom value would be z
equals one minus y.
00:46:51.000 --> 00:46:56.000
The top value is on the sphere.
So, the sphere corresponds to z
00:46:56.000 --> 00:47:01.000
equals square root of one minus
x squared minus y squared.
00:47:01.000 --> 00:47:05.000
So, you'd go from the plane to
the sphere.
00:47:05.000 --> 00:47:09.000
And then, to find the bounds
for x and y, you have to figure
00:47:09.000 --> 00:47:13.000
out what exactly,
what the heck is this region
00:47:13.000 --> 00:47:15.000
here?
So, what is this region?
00:47:15.000 --> 00:47:19.000
Well, we have to figure out,
for what values of x and y the
00:47:19.000 --> 00:47:23.000
plane is below the ellipse.
So, the condition is that,
00:47:23.000 --> 00:47:25.000
sorry, the plane is below the
sphere.
00:47:25.000 --> 00:47:31.000
OK, so, that's when the plane
is below the sphere.
00:47:31.000 --> 00:47:37.000
That means one minus y is less
than square root of one minus x
00:47:37.000 --> 00:47:41.000
squared minus y squared.
So, you have to somehow
00:47:41.000 --> 00:47:43.000
manipulate this to extract
something simpler.
00:47:43.000 --> 00:47:47.000
Well, probably the only way to
do it is to square both sides,
00:47:47.000 --> 00:47:51.000
one minus y squared should be
less than one minus x squared
00:47:51.000 --> 00:47:55.000
minus y squared.
And, if you work hard enough,
00:47:55.000 --> 00:47:57.000
you'll find quite an ugly
equation.
00:47:57.000 --> 00:48:00.000
But, you can figure out what
are, then, the bounds for x
00:48:00.000 --> 00:48:03.000
given y, and then set up the
integral?
00:48:03.000 --> 00:48:06.000
So, just to give you a hint,
the bounds on y will be zero to
00:48:06.000 --> 00:48:09.000
one.
The bounds on x,
00:48:09.000 --> 00:48:10.000
well, I'm not sure you want to
see them,
00:48:10.000 --> 00:48:14.000
but in case you do,
it will be from negative square
00:48:14.000 --> 00:48:18.000
root of 2y minus 2y squared to
square root of 2y minus 2y
00:48:18.000 --> 00:48:20.000
squared.
So, exercise,
00:48:20.000 --> 00:48:25.000
figure out how I got these by
starting from that.
00:48:25.000 --> 00:48:27.000
Now, of course,
if we just wanted the volume of
00:48:27.000 --> 00:48:28.000
this guy, we wouldn't do it this
way.
00:48:28.000 --> 00:48:31.000
We do symmetry,
and actually we'd rotate the
00:48:31.000 --> 00:48:34.000
thing so that our spherical cap
was actually centered on the z
00:48:34.000 --> 00:48:37.000
axis because that would be a
much easier way to set it up.
00:48:37.000 --> 00:48:39.000
But, depending on what function
we are integrating,
00:48:39.000 --> 00:48:42.000
we can't always do that.