WEBVTT
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JOEL LEWIS: Hi.
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Welcome back to recitation.
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In lecture, you've been learning
about triple integration.
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And I have a
problem here for you
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on computing a volume of a
region using a triple integral.
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So let's look at this.
00:00:19.710 --> 00:00:23.260
So I have a volume and
I'm describing it to you;
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it's the volume inside
the paraboloid z
00:00:27.240 --> 00:00:32.030
equals x squared plus y squared
and bounded by the plane z
00:00:32.030 --> 00:00:33.477
equals 2y.
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So I've drawn a little
picture here for you.
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So this is the paraboloid here.
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And we're just taking
a plane cut of it.
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And so this is
going to slice off
00:00:43.370 --> 00:00:46.430
some chunk of that paraboloid,
and what I want to know
00:00:46.430 --> 00:00:49.110
is, what's the volume
of that piece that gets
00:00:49.110 --> 00:00:50.920
cut off by that plane there?
00:00:50.920 --> 00:00:54.730
So below the plane and
above the paraboloid.
00:00:54.730 --> 00:00:57.780
So, why don't you pause
the video, take some time,
00:00:57.780 --> 00:00:59.189
work out this
problem, come back,
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and we can work on it together.
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I hope you had some
luck with this problem.
00:01:10.872 --> 00:01:12.330
I think it's a bit
of a tricky one,
00:01:12.330 --> 00:01:15.140
so let's start to work
through it together.
00:01:15.140 --> 00:01:20.050
So sometimes you have a
problem with a triple integral.
00:01:20.050 --> 00:01:23.480
And you need to set up
your bounds of integration.
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And sometimes you can look
at it and it's just clear
00:01:25.690 --> 00:01:26.510
what they are.
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If you're integrating over
a cube, life is really easy.
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But in this case, this region
that we want to integrate over
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is kind of more
complicated to understand.
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Right?
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So it's easy to see-- well,
relatively easy to see--
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what the bounds on z are.
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So let me draw a couple of
two-dimensional pictures.
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So I'm going to draw the
yz-plane cross section here.
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So in the yz-plane
cross section,
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this paraboloid just
becomes a parabola.
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So that becomes the
parabola z equals
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y squared, which is a plane
section of the paraboloid z
00:02:08.910 --> 00:02:11.710
equals x squared plus y squared.
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And this plane z equals 2y
becomes the line z equals 2y.
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And this little sliver
is a plane section
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of the region in question.
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So we see that z is going from
the paraboloid to the plane.
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And over here, we see that z
is going from the paraboloid
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to the plane.
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But what we really
need to understand then
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is what the relationship
between x and y is.
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So what is the shadow
of this region?
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How are x and y
related to each other?
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How can we bound x in terms
of y or y in terms of x?
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Or should we use cylindrical
coordinates or what?
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And so in order to
that what we need to do
00:02:51.380 --> 00:02:54.870
is we need to figure out-- when
you project this region down,
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when you flatten it along z,
so you're disregarding z now,
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and then you're just looking
at its shadow, its footprint
00:03:01.040 --> 00:03:04.707
in the xy-plane-- you
want to figure out,
00:03:04.707 --> 00:03:05.540
what is that region?
00:03:05.540 --> 00:03:06.498
What does it look like?
00:03:06.498 --> 00:03:13.670
So somehow we'll project down
and there will be some region R
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down here.
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So I'll call this region
R. And that region
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will be the projection of
this solid region down.
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And it has some
boundary curve-- C,
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say-- the boundary
curve of the region
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R. Just in case we need
to refer to them later,
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it's good to give them letters
so that they have names.
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So what we need to figure out
now is what is this region R?
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Now this is tough to do by
just intuitive reasoning
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or just by looking at
this picture I've drawn.
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So in this case, we're kind
of forced to use some algebra.
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All right.
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So what do we know about this
region R and this curve C?
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Well, C is the
projection downwards
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of the curve of
intersection of this plane
00:04:06.470 --> 00:04:08.830
with this paraboloid, right?
00:04:08.830 --> 00:04:12.140
So it's the projection down
at this curve intersection.
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So what does that mean
about its equation?
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Well, it means it's what we
get if we solve for z in one
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of the two equations
of the surfaces
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and plug it into the other.
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And that will give us an
equation with just x and y,
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and that will be the
equation of this curve C. OK.
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So in our case, that means that
C is given by this equation
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x squared plus y
squared equals 2y.
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All right.
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So whenever x squared
plus y squared equals
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2y, that's a point (x, y) such
that directly above that point
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is a place where the plane
intersects the paraboloid.
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Well, OK.
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So what is this curve?
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Well, a little bit of algebra
can help us sort that out.
00:05:01.910 --> 00:05:05.230
If you bring the 2y over
here and complete the square,
00:05:05.230 --> 00:05:06.790
you can see that
we can rewrite this
00:05:06.790 --> 00:05:12.530
as x squared plus y
minus 1 squared equals 1.
00:05:12.530 --> 00:05:15.320
I brought the 2y over,
I've added 1 to both sides,
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and I've factored the y part.
00:05:19.920 --> 00:05:23.065
And so this is an easy
equation to recognize.
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This is the equation of a
circle with center (0, 1)
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and radius 1.
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So let's draw that.
00:05:29.200 --> 00:05:32.480
And so here is a picture
of what the shadow looks
00:05:32.480 --> 00:05:35.790
like in the xy-plane.
00:05:35.790 --> 00:05:42.357
So the center is at height
1, and then it's this circle.
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That's almost a circle.
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It looks enough like a
circle for my purposes.
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So this is the region R.
It's a circular region
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of radius 1 with center (0, 1).
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OK, great.
00:05:54.090 --> 00:05:55.847
So I'm just going to
shade that in again
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because I like doing that.
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OK.
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So that's the region R.
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So what is this region R?
00:06:04.760 --> 00:06:05.990
Let's look back over here.
00:06:05.990 --> 00:06:09.860
It's the shadow of our solid
region in the xy-plane.
00:06:09.860 --> 00:06:12.430
So when you project down,
that's the region that you get.
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So why do we need that?
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So we know when we set up
this triple integral, z is
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going to be going from the
paraboloid up to the plane.
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That's going to be the
innermost integral,
00:06:25.440 --> 00:06:27.125
but then the middle
integral is going
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to be y in terms of
x or x in terms of y.
00:06:30.050 --> 00:06:33.550
Or if we do polar coordinates
or cylindrical coordinates,
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it's going to be R
in terms of theta.
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So we need to figure out
what the boundary is,
00:06:39.530 --> 00:06:43.500
what that region looks like
over which we'll be integrating
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for the outer two integrals.
00:06:46.644 --> 00:06:48.810
OK, so now I've been saying
we could use cylindrical
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or we could use rectangular.
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What do we want to use?
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Well, so this is a circle.
00:06:53.640 --> 00:06:55.970
It's not centered at
the origin, but it
00:06:55.970 --> 00:07:00.310
is tangent to one of
the axes at the origin.
00:07:00.310 --> 00:07:04.110
So this is a reasonably
nice situation
00:07:04.110 --> 00:07:10.210
to do polar coordinates in,
or cylindrical coordinates.
00:07:10.210 --> 00:07:12.070
You have to remember
from when you learned
00:07:12.070 --> 00:07:14.000
cylindrical and polar
coordinates what
00:07:14.000 --> 00:07:16.190
the equation of
such a circle is.
00:07:16.190 --> 00:07:18.550
And so I'm going to
write it down here,
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and I'm going to invite you
to go look up why this is true
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if you don't remember.
00:07:25.520 --> 00:07:29.330
This curve has
equation in polar--
00:07:29.330 --> 00:07:34.600
these are the x- and y-axes
here-- so this curve has,
00:07:34.600 --> 00:07:41.871
in polar coordinates, the
equation r equals 2 sine theta.
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All right.
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So that gives me
this curve here.
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The outer boundary.
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And now what I want is, I
don't just want the curve.
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I want to integrate
over the whole region,
00:07:53.070 --> 00:07:54.620
and I want to
integrate over it once.
00:07:54.620 --> 00:07:56.578
Remember, polar coordinates
are a little tricky
00:07:56.578 --> 00:07:59.970
because you have to worry about
are you overlapping and so on.
00:07:59.970 --> 00:08:01.340
So how does this work?
00:08:01.340 --> 00:08:04.270
At theta equals 0,
or at the origin,
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and then as theta grows, we
get further and further away.
00:08:08.820 --> 00:08:13.780
So this is our
radius growing out.
00:08:13.780 --> 00:08:16.560
And then at pi over 2, we're
at the top point of the circle.
00:08:16.560 --> 00:08:18.970
And then as it comes back
into pi, it comes back in.
00:08:18.970 --> 00:08:22.410
So we want theta
going from 0 less than
00:08:22.410 --> 00:08:25.770
or equal to theta less
than or equal to pi here.
00:08:25.770 --> 00:08:27.710
So at pi over 2 at
the top, and at pi
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it comes back for
the first time.
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And what about r?
00:08:30.810 --> 00:08:33.730
Well, it looks like r
has to go all the way out
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to 2 sine theta.
00:08:34.450 --> 00:08:36.840
And in fact, we always want
it to start at the origin.
00:08:36.840 --> 00:08:44.120
So we always want r to go from
0 to this outer boundary, 2
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sine theta.
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So this describes
this region big R
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that we're trying
to integrate over.
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This circular region
in polar coordinates.
00:08:55.210 --> 00:08:55.860
So OK.
00:08:55.860 --> 00:08:57.505
So it's a fairly
easy description
00:08:57.505 --> 00:08:58.380
in polar coordinates.
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You could also describe it
in rectangular coordinates,
00:09:00.630 --> 00:09:03.770
and you could try to solve
the problem that way.
00:09:03.770 --> 00:09:06.577
I'm not going to do it for you,
but you could give it a shot
00:09:06.577 --> 00:09:09.035
and see if you can come out
with the same answer in the end
00:09:09.035 --> 00:09:12.050
that we do.
00:09:12.050 --> 00:09:13.300
So OK.
00:09:13.300 --> 00:09:14.710
So now, what have we done?
00:09:14.710 --> 00:09:17.070
Well, I haven't
written our bounds,
00:09:17.070 --> 00:09:19.260
so let me write our
bounds on z right here.
00:09:19.260 --> 00:09:22.580
So we know that z is
going from the paraboloid.
00:09:25.120 --> 00:09:27.240
If we look, it's
the paraboloid z
00:09:27.240 --> 00:09:29.800
equals x squared plus
y squared-- but we're
00:09:29.800 --> 00:09:32.080
working in cylindrical
coordinates now,
00:09:32.080 --> 00:09:34.970
so we need to write this
in terms of r and theta--
00:09:34.970 --> 00:09:39.100
so that's z is going
from r squared,
00:09:39.100 --> 00:09:44.100
and it's going up to the
plane z equals 2y-- now y
00:09:44.100 --> 00:09:46.690
in cylindrical coordinates
is r sine theta.
00:09:46.690 --> 00:09:51.160
So z is going from r
squared to 2r sine theta.
00:09:51.160 --> 00:09:52.970
So let's go write
that down over here.
00:09:52.970 --> 00:10:01.260
So z is going from-- just ignore
that-- from r squared less than
00:10:01.260 --> 00:10:03.590
or equal to z, and
it's going all the way
00:10:03.590 --> 00:10:07.560
up to 2r sine theta.
00:10:07.560 --> 00:10:12.170
So these three equations
describe our region.
00:10:12.170 --> 00:10:13.150
Yeah?
00:10:13.150 --> 00:10:15.621
0 less than theta less than
pi: that just says theta.
00:10:15.621 --> 00:10:16.120
OK?
00:10:16.120 --> 00:10:18.400
Then when theta is
going from 0 to pi--
00:10:18.400 --> 00:10:22.870
r going from 0 to 2 sine theta--
that says in the xy-plane,
00:10:22.870 --> 00:10:25.410
we're tracing out
this circular shadow.
00:10:25.410 --> 00:10:28.960
And then as z goes from r
squared to 2r sine theta, that
00:10:28.960 --> 00:10:32.380
says above this shadow
we're above the paraboloid
00:10:32.380 --> 00:10:33.660
and below the plane.
00:10:33.660 --> 00:10:36.631
So that's exactly the
region that we want.
00:10:36.631 --> 00:10:37.130
So OK.
00:10:37.130 --> 00:10:40.840
So now, how do we get its volume
after we figured this out?
00:10:40.840 --> 00:10:44.580
Well, we write down
the triple integral.
00:10:44.580 --> 00:10:49.330
So V, the volume
of a region D, is
00:10:49.330 --> 00:10:55.800
equal to the triple integral
over that solid of dV.
00:10:55.800 --> 00:10:56.400
OK?
00:10:56.400 --> 00:10:59.370
And in our case, in
cylindrical coordinates,
00:10:59.370 --> 00:11:08.540
dV is going to be dz times r
dr d theta, or r dz dr d theta.
00:11:08.540 --> 00:11:09.040
OK?
00:11:09.040 --> 00:11:21.030
So this is equal to, if we're
integrating, r dz dr d theta.
00:11:21.030 --> 00:11:23.840
And now we need to
put in our bounds.
00:11:23.840 --> 00:11:26.340
If we look over on this
side of me, here they are.
00:11:26.340 --> 00:11:28.640
And these are our bounds
that we're going to be using.
00:11:28.640 --> 00:11:33.400
So theta is going from 0 to pi.
00:11:33.400 --> 00:11:40.650
And r is going from
0 to 2 sine theta.
00:11:40.650 --> 00:11:48.200
And z is going from r
squared to 2r sine theta.
00:11:50.780 --> 00:11:54.100
So this triple integral
gives us precisely the volume
00:11:54.100 --> 00:11:55.720
of our region.
00:11:55.720 --> 00:11:58.110
And in order to figure
out what that volume is,
00:11:58.110 --> 00:12:00.260
we just have to
evaluate this integral.
00:12:00.260 --> 00:12:01.960
So let's start doing that.
00:12:01.960 --> 00:12:05.160
I don't think I'm going
to go quite all the way,
00:12:05.160 --> 00:12:07.680
but I'll do most of the work.
00:12:07.680 --> 00:12:08.260
So OK.
00:12:08.260 --> 00:12:10.110
So let's do the
innermost integral first.
00:12:10.110 --> 00:12:11.620
Whenever you have
a triple integral
00:12:11.620 --> 00:12:13.290
like this-- a nice
iterated integral--
00:12:13.290 --> 00:12:15.670
you always start at the
inside and work your way out.
00:12:15.670 --> 00:12:18.560
Well here, our integrand
is r, and we're
00:12:18.560 --> 00:12:20.250
integrating with
respect to z-- and r
00:12:20.250 --> 00:12:22.830
doesn't have any z's in it--
so this inner integral is
00:12:22.830 --> 00:12:24.690
going to be easy.
00:12:24.690 --> 00:12:28.730
So I'm going to rewrite
this as equal to-- we keep
00:12:28.730 --> 00:12:31.696
our outer two bounds, so
it's still from 0 to pi,
00:12:31.696 --> 00:12:42.980
and it's still from 0 to 2
sine theta-- of 2r squared sine
00:12:42.980 --> 00:12:48.460
theta minus r cubed dr d theta.
00:12:48.460 --> 00:12:51.760
So what I've done here
is I've just integrated.
00:12:51.760 --> 00:12:56.516
I get the anti-derivative
of r dz is r*z.
00:12:56.516 --> 00:12:57.890
And so then I take
the difference
00:12:57.890 --> 00:12:58.931
between those two bounds.
00:12:58.931 --> 00:13:03.130
So I get r times 2r sine
theta minus r times r squared.
00:13:03.130 --> 00:13:07.190
So r times 2r sine theta
is 2r squared sine theta.
00:13:07.190 --> 00:13:09.052
Minus r times r squared
is minus r cubed.
00:13:09.052 --> 00:13:10.760
OK, so I've just done
the first integral.
00:13:10.760 --> 00:13:13.770
So now integrating
with respect to r.
00:13:13.770 --> 00:13:16.490
OK, this second one
isn't so bad either.
00:13:16.490 --> 00:13:19.240
As far as r is concerned,
this is just a polynomial.
00:13:19.240 --> 00:13:22.100
Theta is constant with
respect to r when we're
00:13:22.100 --> 00:13:23.320
doing an integral like this.
00:13:23.320 --> 00:13:23.980
So OK.
00:13:23.980 --> 00:13:26.140
So the second integral
is not too bad either.
00:13:26.140 --> 00:13:30.450
So this is the integral-- so
our outer integral from 0 to pi
00:13:30.450 --> 00:13:33.540
sticks around-- let's
not do this one in one
00:13:33.540 --> 00:13:35.715
fell swoop I think--
so it's going
00:13:35.715 --> 00:13:42.900
to become 2 r cubed
over 3, sine theta,
00:13:42.900 --> 00:13:46.360
minus r to the fourth over 4.
00:13:46.360 --> 00:13:49.290
And we're taking
that between r equals
00:13:49.290 --> 00:13:55.420
0 and r equals 2 sine theta.
00:13:55.420 --> 00:13:59.300
And then that whole thing is
going to be integrated d theta.
00:13:59.300 --> 00:14:01.160
So what do we get
when we plug this in?
00:14:01.160 --> 00:14:04.770
Well, at r equals zero, this
is just 0, so that's easy.
00:14:04.770 --> 00:14:08.090
And so we need the top
one, r equals 2 sine theta.
00:14:08.090 --> 00:14:11.850
So this is going to give
me something like 16/3 sine
00:14:11.850 --> 00:14:15.930
to the fourth theta minus
4 sine to the fourth theta,
00:14:15.930 --> 00:14:27.830
so I think that works out to
be 4/3 sine to the fourth theta
00:14:27.830 --> 00:14:30.250
d theta, between 0 and pi.
00:14:30.250 --> 00:14:33.950
So now you have to remember
how to do integrals like this.
00:14:33.950 --> 00:14:36.990
So this is something you
probably learned back
00:14:36.990 --> 00:14:44.170
in the trig integral section of
your Calculus I or 18.01 class.
00:14:44.170 --> 00:14:45.950
So when it's an
even power here, I
00:14:45.950 --> 00:14:50.110
think the thing that we do is
we use our half-angle formulas.
00:14:50.110 --> 00:14:55.040
So now I'm going to tell you
what your final steps are.
00:14:55.040 --> 00:14:58.310
So first, you're going to
use your half-angle formula.
00:14:58.310 --> 00:15:00.060
So what is that
half-angle formula?
00:15:00.060 --> 00:15:07.850
So it's sine squared theta
is equal to 1 minus cosine
00:15:07.850 --> 00:15:11.410
2 theta over 2.
00:15:11.410 --> 00:15:13.600
So you're going to have to
plug this in here, right?
00:15:13.600 --> 00:15:16.900
Sine to the fourth is sine
squared quantity squared.
00:15:16.900 --> 00:15:20.030
And then you're going to have
a cosine squared 2 theta,
00:15:20.030 --> 00:15:22.587
so you're going to have to
use the double-angle formula.
00:15:22.587 --> 00:15:25.170
This time you're going to have
to use the double-angle formula
00:15:25.170 --> 00:15:27.640
for cosine, which is very
similar, although not
00:15:27.640 --> 00:15:28.440
exactly the same.
00:15:28.440 --> 00:15:30.570
So you're going to
have to use those two
00:15:30.570 --> 00:15:31.646
double-angle formulas.
00:15:31.646 --> 00:15:33.020
After that, you'll
have something
00:15:33.020 --> 00:15:35.549
that is straightforward
to integrate.
00:15:35.549 --> 00:15:38.090
So you'll have something that's
straightforward to integrate.
00:15:38.090 --> 00:15:40.210
You'll integrate
it, and if I'm not
00:15:40.210 --> 00:15:45.360
mistaken, what
you get at the end
00:15:45.360 --> 00:15:49.960
is that you just get a fairly
nice and simple pi over 2
00:15:49.960 --> 00:15:51.040
as your answer.
00:15:51.040 --> 00:15:54.150
So you can check your
work there, and make sure
00:15:54.150 --> 00:15:56.210
that you've got out
pi over 2 at the end.
00:15:56.210 --> 00:15:58.400
And hopefully, if
you tried to do
00:15:58.400 --> 00:16:00.590
this using rectangular
coordinates,
00:16:00.590 --> 00:16:03.040
you also came out with
something like this as well.
00:16:03.040 --> 00:16:05.450
In that case, you would have
to do a trig substitution
00:16:05.450 --> 00:16:07.790
at some point to
evaluate your intervals,
00:16:07.790 --> 00:16:10.330
or you might have
an arcsine involved.
00:16:10.330 --> 00:16:13.230
Something like that will happen.
00:16:13.230 --> 00:16:15.380
But it should also give
you pi over 2, of course.
00:16:15.380 --> 00:16:17.380
Because it's the
same region, just
00:16:17.380 --> 00:16:18.970
described in a different way.
00:16:18.970 --> 00:16:21.930
So let me quickly
recap what we did.
00:16:21.930 --> 00:16:27.200
Way back over here, we had this
description of this region.
00:16:27.200 --> 00:16:32.310
So it was the region above our
paraboloid and below a plane.
00:16:32.310 --> 00:16:35.069
And so when we're
setting this up,
00:16:35.069 --> 00:16:37.360
we have to figure out, in
order to do a triple integral
00:16:37.360 --> 00:16:41.774
over this region, in
order to find its volume,
00:16:41.774 --> 00:16:43.440
we have to pick an
order of integration,
00:16:43.440 --> 00:16:46.510
and then we have to
know what the bounds are
00:16:46.510 --> 00:16:49.610
for the inside in terms of
the outer two variables,
00:16:49.610 --> 00:16:52.370
for the middle one in terms of
the outermost one, and so on.
00:16:52.370 --> 00:16:56.650
So in this case,
that means-- First,
00:16:56.650 --> 00:17:01.120
it was a natural choice to
make z the first variable--
00:17:01.120 --> 00:17:02.605
the innermost variable.
00:17:02.605 --> 00:17:03.980
And so then after
that, we needed
00:17:03.980 --> 00:17:08.220
to project to find the
relationship in the xy-plane
00:17:08.220 --> 00:17:09.580
between the other variables.
00:17:09.580 --> 00:17:17.230
Now in this case, we did that
by solving this little algebra
00:17:17.230 --> 00:17:17.780
problem here.
00:17:17.780 --> 00:17:23.151
We solved for z in the two
surfaces that we were given,
00:17:23.151 --> 00:17:24.650
and we set them
equal to each other.
00:17:24.650 --> 00:17:27.080
And so this gives us a
description for the boundary
00:17:27.080 --> 00:17:28.310
curve for our region.
00:17:28.310 --> 00:17:31.550
And because it's a
nice circle, this
00:17:31.550 --> 00:17:34.630
suggested that one possibility
was cylindrical coordinates.
00:17:34.630 --> 00:17:38.090
So we went ahead, and we found
in cylindrical coordinates
00:17:38.090 --> 00:17:39.840
the description of this shadow.
00:17:39.840 --> 00:17:42.940
And then we used the
knowledge we previously
00:17:42.940 --> 00:17:46.140
had to describe the whole region
in cylindrical coordinates.
00:17:46.140 --> 00:17:48.550
So we had this description
of our entire region.
00:17:48.550 --> 00:17:50.320
And then to compute
its volume, we just
00:17:50.320 --> 00:17:52.200
set up the triple
integral volume
00:17:52.200 --> 00:17:55.330
is equal to a
triple integral dV.
00:17:55.330 --> 00:17:57.170
In our case, dV-- since
we're in cylindrical
00:17:57.170 --> 00:17:59.890
coordinates-- that's
r dz dr d theta.
00:17:59.890 --> 00:18:04.240
We put in our bounds, and then
we evaluated the integral.
00:18:04.240 --> 00:18:05.854
I'll stop there.