WEBVTT

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CHRISTINE BREINER: Welcome
back to recitation.

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In this video, we want to
work on using the change

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of variables technique.

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In particular, we're going to
look at the following problem.

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It says, using the change
of variables u is equal to x

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squared minus y squared and
v is equal to y divided by x,

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supply the limits and integrand
for the following integral,

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which is the double integral
over region R of 1 over x

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squared, dx*dy.

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And R is the infinite region
in the first quadrant that

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is both under the
curve y equals 1

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over x, and to the
right of the curve

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x squared minus y
squared equals 1.

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So this is a
challenging problem.

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Again, I want to
point out we just

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want to find the limits
and the integrand.

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You don't actually have
to compute the integral.

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But it is going to be
tough, but stick with it.

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Pause the video, give it
your best shot-- hopefully

00:00:57.870 --> 00:01:00.900
you find the appropriate
limits and integrand--

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and then when you
feel comfortable,

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bring the video back up, and
I'll show you how I do it.

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OK.

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Welcome back.

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So once again,
what we want to do

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is this change of
variables problem

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where we've defined u to be
x squared minus y squared,

00:01:20.750 --> 00:01:24.840
v to be y divided by x, and
we have this region that

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is in the first
quadrant and it's

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under the curve y
equals 1 divided by x

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and it's to the right of the
curve x squared minus y squared

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equals 1.

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And we want to
compute the limits

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and integrand for that
particular integral.

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So what I'm going to
do, to try and make

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this as organized
as possible, is

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I'm going to try first
to graph the region R,

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or to figure out what the
region R is in the xy-plane.

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Then I'm going to try and
figure out what the region

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R is mapped to in the uv-plane.

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So what it looks
like in the uv-plane.

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That will give me my limits.

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And then I'm going to try
and determine the Jacobian.

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And then I will determine
from that and the fact

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that I started with 1 divided by
x squared as my function I was

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integrating, I will
put those two together

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to figure out the integrand.

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So there are a bunch of
steps to these problems.

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But the first one, again is I'm
going to graph the region R.

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So I'm going to give you a
very rough sketch, over here,

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of the region R. And I know
it's in the first quadrant

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and I know it's infinite.

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I was already told that.

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OK.

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So in the xy-plane, the region
R is below the curve y equals 1

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over x.

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So let me draw that curve.

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Again, this is very rough.

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This is a rough sketch.

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I'm putting up no
scale on purpose.

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I'll put in one value,
maybe, in this whole thing.

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OK?

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And so this is the curve
y equals one over x.

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And then I need
the curve which is

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part of the hyperbola that
is x squared minus y squared

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equals 1.

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So I'll draw in
the part we need,

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which looks roughly like this.

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Something like that.

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Again, this is not meant
to be an exact graph,

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but to give you an idea of
what the region looks like.

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And the only thing
I'm going to mention

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is that this point we know
is x equals 1 and y equals 0.

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So the region we're
interested in that

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is both to the right of x
squared minus y squared equals

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1 and below y equals 1 over
x and in the first quadrant

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is exactly this region
I'm shading here.

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So we want to understand
what the values of u and v

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are along these bounds.

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We need to understand
where this region maps

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to when I do the
change of variables

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in order to understand
what the limits are.

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So let me put the graph of
this region in the uv-plane

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so that we can really
understand what our bounds are.

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And I know already
where it's going.

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So I'm going to just make
the first quadrant, because I

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know this is going into
the first quadrant.

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So it doesn't always
work that something

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in the first quadrant maps
into the first quadrant,

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but in this case, I already did
the work, so I know it does.

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So let me point out a few things
about where this region R maps.

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The first thing I
want to point out

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is that we actually
know that this curve,

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under the change of
variables, maps to u equals 1.

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Because if you remember, u
is equal to x squared minus y

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squared.

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So this whole curve is
going to map to u equals 1.

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Now, I don't want the
whole curve for my region.

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I only want this
little piece of it.

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So I'm going to
have-- in my uv plane,

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I'm going to have
some segment at 1.

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And actually, I'll
just know that it's

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some part of the line u equals
1 is going to show up in there.

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But if you notice, I know
where it starts right away.

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Because at x equal
1, y equals 0,

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if I look at what v is-- if
we come back here and remember

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what v is-- at x equal
1, y equals 0-- v is 0.

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And so my starting point on this
segment-- if we come back here,

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my starting point
on this segment

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is actually also at (1, 0).

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OK?

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So I know there's
some point right

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here that maps down to here
where the segment will stop.

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I'll find that point
later, algebraically.

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Right?

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And then now we need to figure
out where these two curves go.

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And then we can get
a picture, and then

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we'll figure out
what that point is,

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and we'll understand
all the limits.

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So the first thing I want to
point out is along this curve,

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we have y equals 0
and x is non-zero.

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And just to help
ourselves, I'm going

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to rewrite what the
change of variables

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is here, so I don't have to keep
walking over to the other side.

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Our change of variables
was u is equal to x

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squared minus y squared, and
v was equal to y divided by x.

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So this whole curve
has y equals 0.

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So what happens to u and what
happens to v along that curve?

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Well, u is going
to be x squared,

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and v is going to equal 0.

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And so the point
of this, really,

00:06:18.710 --> 00:06:22.520
is that even though
in u, this curve maybe

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is mapping at a different speed
in some form to this curve

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here, it's still-- it's just
taking that segment goes--

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or that infinitely long ray goes
to an infinitely long ray here

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along the u-axis.

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And again, that's because
along this ray, y equals 0.

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And so v is equal
to 0 everywhere on

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that ray and u is positive--
it's equal to x squared.

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OK?

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So I'm going to move
the u out of the way,

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because we're going to say
this is part of the region,

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or that's one bound
of the region.

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And now I have to figure
out where this curve goes.

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This curve is slightly
more complicated,

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but I can still figure it out.

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So I'm going to show you how I
do that sort of algebraically.

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That curve-- if you
notice, if you remember--

00:07:04.590 --> 00:07:07.580
is y equals 1 divided by x.

00:07:07.580 --> 00:07:10.550
So that means that
on that curve--

00:07:10.550 --> 00:07:14.740
let me even write it down--
on y equals 1 divided by x, v

00:07:14.740 --> 00:07:17.710
is equal to 1 divided
by x divided by x.

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So v is equal to 1 divided
by x squared, right?

00:07:23.470 --> 00:07:26.500
And then what does
that mean about u?

00:07:26.500 --> 00:07:33.620
u, then, is equal to-- well,
x squared is 1 divided by v,

00:07:33.620 --> 00:07:39.140
and then y squared, because y
squared on that curve is just 1

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divided by x squared,
is v. So let me just

00:07:45.130 --> 00:07:47.100
make sure we all followed
that one more time.

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We're looking at where
the curve y equals 1

00:07:49.260 --> 00:07:51.890
over x goes in the change
of variables, right?

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So that's the top curve up here.

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y equals 1 over x is the
top curve of our region R.

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So we want to know
where that goes.

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Well, on y equals 1
over x, v is exactly

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equal to 1 over x squared,
because v-- we know--

00:08:03.690 --> 00:08:04.510
is y over x.

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So if I just substitute in
for y, I get 1 over x squared.

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Now, if I look at
this relationship,

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this means x squared is equal
to 1 over v. So in terms of u,

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x squared becomes 1
over v. And then y

00:08:17.400 --> 00:08:20.120
squared-- which is
1 over x squared--

00:08:20.120 --> 00:08:25.470
become v. So that curve is
u equals 1 over v minus v.

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Now that curve,
roughly, is going

00:08:27.434 --> 00:08:28.600
to look something like this.

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And it might seem strange.

00:08:36.950 --> 00:08:39.400
The thing is, I'm graphing
this in the uv-plane,

00:08:39.400 --> 00:08:42.430
and I'm writing what looks
like u as a function of v,

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and so it's sort of turned
around from how you usually

00:08:44.870 --> 00:08:46.070
see these things written.

00:08:46.070 --> 00:08:51.480
But this is the equation
that describes this curve.

00:08:51.480 --> 00:08:53.300
And that is sufficient
to understand,

00:08:53.300 --> 00:08:57.500
because when we use our--
when we determine our bounds,

00:08:57.500 --> 00:09:01.190
we can determine our bounds from
u equals 0 now, to u equals 1

00:09:01.190 --> 00:09:04.700
over v minus v. So we
now have the bounds in u.

00:09:04.700 --> 00:09:06.630
We're actually doing quite well.

00:09:06.630 --> 00:09:08.830
So we have this region.

00:09:08.830 --> 00:09:11.430
We now have the bounds
completely in u.

00:09:11.430 --> 00:09:16.100
u is going from u equals 0
to u equals 1 over v minus v.

00:09:16.100 --> 00:09:19.090
But the problem is now
we don't know the bounds

00:09:19.090 --> 00:09:22.690
in v. We don't know what
the bounds are in v,

00:09:22.690 --> 00:09:26.590
and so we have to be
a little bit careful.

00:09:26.590 --> 00:09:27.387
So actually, no.

00:09:27.387 --> 00:09:28.220
I think I was wrong.

00:09:28.220 --> 00:09:29.230
It's not 0, is it?

00:09:29.230 --> 00:09:31.630
I said that twice now,
and that was incorrect.

00:09:31.630 --> 00:09:36.220
u is going from 1, to 1 over
v minus v. So I apologize.

00:09:36.220 --> 00:09:39.360
Because the slices of u
are going from whatever

00:09:39.360 --> 00:09:41.480
the u-value starts--
which is at the value 1--

00:09:41.480 --> 00:09:42.780
and it's coming this way.

00:09:42.780 --> 00:09:44.720
So I apologize.

00:09:44.720 --> 00:09:47.470
I was moving my arm like
I was doing the v-values,

00:09:47.470 --> 00:09:50.040
but I actually wanted
to do the u-values.

00:09:50.040 --> 00:09:52.340
So I want to go from where
u starts-- which is at u

00:09:52.340 --> 00:09:57.030
equals 1-- to where u stops--
which is when it hits the curve

00:09:57.030 --> 00:09:59.440
1 over v minus v equals u.

00:09:59.440 --> 00:10:01.540
So hopefully I didn't
confuse anyone by that.

00:10:01.540 --> 00:10:03.860
I'm glad I caught it, then.

00:10:03.860 --> 00:10:06.406
OK, so now we understand
the bounds in u.

00:10:06.406 --> 00:10:07.780
And then to
understand the bounds

00:10:07.780 --> 00:10:09.570
in v, all we need to
understand is what

00:10:09.570 --> 00:10:12.850
is the v-value at this point.

00:10:12.850 --> 00:10:15.940
So once I know the
v-value at this point,

00:10:15.940 --> 00:10:18.010
then I'm done with the bounds.

00:10:18.010 --> 00:10:20.220
So let's see if
we can find that.

00:10:20.220 --> 00:10:24.500
Well, the v-value at that point
is going to be at the point

00:10:24.500 --> 00:10:26.640
where these two
curves intersect.

00:10:26.640 --> 00:10:32.019
So let's see if we can do a
little algebra to understand

00:10:32.019 --> 00:10:33.060
what that will look like.

00:10:33.060 --> 00:10:35.660
So let me point out that
where those curves intersect,

00:10:35.660 --> 00:10:42.250
I have the equation x squared
minus 1 over x squared

00:10:42.250 --> 00:10:44.130
is equal to 1.

00:10:44.130 --> 00:10:46.980
And if I want to find
x-values that satisfy this,

00:10:46.980 --> 00:10:49.836
I'm also looking for
x-values that satisfy

00:10:49.836 --> 00:10:53.580
x to the fourth minus 1
is equal to x squared,

00:10:53.580 --> 00:10:57.900
which I can rewrite as x to the
fourth minus x squared minus 1

00:10:57.900 --> 00:10:59.790
is equal to 0.

00:10:59.790 --> 00:11:04.010
So I can actually use
the quadratic formula

00:11:04.010 --> 00:11:07.350
on this in terms of x squared.

00:11:07.350 --> 00:11:12.890
So what I get is I get x
squared is equal to 1--

00:11:12.890 --> 00:11:16.890
I get plus or minus
root 5-- over 2.

00:11:16.890 --> 00:11:18.947
And if you look at it,
the one you're actually

00:11:18.947 --> 00:11:21.280
interested in-- you can figure
this out pretty quickly--

00:11:21.280 --> 00:11:23.340
is the one that is plus.

00:11:23.340 --> 00:11:24.110
OK?

00:11:24.110 --> 00:11:27.080
I want the one that
is plus root 5 over 2.

00:11:27.080 --> 00:11:33.090
So then that means x is the
square root of this quantity

00:11:33.090 --> 00:11:35.740
at that point, right?

00:11:35.740 --> 00:11:37.600
Or I could actually
think about it this way.

00:11:37.600 --> 00:11:39.300
Let me point out
this. v is equal to 1

00:11:39.300 --> 00:11:42.330
over x squared at that
point, because it's

00:11:42.330 --> 00:11:44.494
on that curve where we
were talking about y

00:11:44.494 --> 00:11:45.160
equals 1 over x.

00:11:45.160 --> 00:11:46.940
So v is 1 over x squared.

00:11:46.940 --> 00:11:51.530
So 1 over x squared is
just 1 over this quantity.

00:11:51.530 --> 00:11:52.950
So it's the reciprocal of this.

00:11:52.950 --> 00:11:57.640
It's also negative 1
plus root 5, over 2.

00:11:57.640 --> 00:11:59.460
You can check that
if you need to.

00:11:59.460 --> 00:12:02.200
But I will write it down
this way as the following:

00:12:02.200 --> 00:12:05.790
this is the point 1 comma a.

00:12:05.790 --> 00:12:12.530
And if I come over here, I will
denote a will equal negative 1

00:12:12.530 --> 00:12:15.480
plus root 5, over 2.

00:12:15.480 --> 00:12:17.810
And that's really just
1 divided by x squared.

00:12:17.810 --> 00:12:20.620
So let me point that
out again, that a

00:12:20.620 --> 00:12:24.987
is equal to 1
divided by x squared

00:12:24.987 --> 00:12:26.195
at the point of intersection.

00:12:32.220 --> 00:12:34.540
So hopefully you
can see all that.

00:12:34.540 --> 00:12:38.410
So that tells us our
bounds completely.

00:12:38.410 --> 00:12:40.480
We still have some work to do.

00:12:40.480 --> 00:12:42.020
So I'm going to
put in the bounds

00:12:42.020 --> 00:12:44.259
and I'm going to
leave an empty space.

00:12:44.259 --> 00:12:44.800
Actually, no.

00:12:44.800 --> 00:12:46.652
I won't do that,
because this can get

00:12:46.652 --> 00:12:47.860
a little messy and confusing.

00:12:47.860 --> 00:12:49.402
So I'm just going
to do the Jacobian,

00:12:49.402 --> 00:12:51.610
and then we'll figure it
all out and write the answer

00:12:51.610 --> 00:12:53.760
right at the end, so
there's no confusion.

00:12:53.760 --> 00:12:56.410
But hopefully you see at this
point that we have the bounds.

00:12:56.410 --> 00:12:59.830
We know that u goes from
1, to 1 over v minus v.

00:12:59.830 --> 00:13:02.910
And v goes from 0
up to a, where a

00:13:02.910 --> 00:13:05.400
is the value I've written here.

00:13:05.400 --> 00:13:07.210
So we know the bounds.

00:13:07.210 --> 00:13:10.250
So now we have to figure
out the integrand.

00:13:10.250 --> 00:13:14.350
So let's first compute
the Jacobian, OK?

00:13:14.350 --> 00:13:19.940
So now we're looking
at del u, v del x, y,

00:13:19.940 --> 00:13:23.410
using the notation
we've seen in class.

00:13:23.410 --> 00:13:25.602
And so del u, v
del x, y is going

00:13:25.602 --> 00:13:28.680
to be the determinant
of the following matrix:

00:13:28.680 --> 00:13:30.690
2x, negative 2y.

00:13:30.690 --> 00:13:33.130
And then the derivative
respect to v of x

00:13:33.130 --> 00:13:36.340
is negative y over x squared.

00:13:36.340 --> 00:13:39.810
And the derivative of v with
respect to y is just 1 over x.

00:13:39.810 --> 00:13:49.850
So if I take the determinant
of that, I get 2 minus 2 y

00:13:49.850 --> 00:13:52.600
squared over x squared.

00:13:52.600 --> 00:13:55.275
Which if you notice, in terms
of our change of variables,

00:13:55.275 --> 00:14:04.370
is exactly equal to 2 minus 2 v
squared, because v is y over x.

00:14:04.370 --> 00:14:06.340
And so I can rewrite
this as 2 times

00:14:06.340 --> 00:14:10.010
the quantity 1 minus v squared.

00:14:10.010 --> 00:14:11.190
OK?

00:14:11.190 --> 00:14:13.910
So, so far so good, hopefully.

00:14:13.910 --> 00:14:18.900
Now let's figure out how
to do the final step.

00:14:18.900 --> 00:14:21.780
So the final step-- I'm
going to come back over

00:14:21.780 --> 00:14:24.795
and just remind us what
the integrand was, OK?

00:14:24.795 --> 00:14:28.230
If we come over here,
we're reminded that we were

00:14:28.230 --> 00:14:33.050
integrating over the region
R of 1 over x squared, dx*dy.

00:14:33.050 --> 00:14:33.550
Right?

00:14:33.550 --> 00:14:35.760
That's what we were
interested in initially.

00:14:35.760 --> 00:14:40.672
So now, if we come back, I'm
going to write that down just

00:14:40.672 --> 00:14:41.755
to have it as a reference.

00:14:48.480 --> 00:14:50.130
OK, that's what
we had initially.

00:14:50.130 --> 00:14:51.220
Let me make sure.

00:14:51.220 --> 00:14:52.950
Yes, that's what
we had initially.

00:14:52.950 --> 00:15:03.030
And so now we know dx*dy is
equal to du*dv over 2 times 1

00:15:03.030 --> 00:15:04.620
minus v squared.

00:15:04.620 --> 00:15:07.730
So that is going to
replace the dx*dy.

00:15:07.730 --> 00:15:10.470
And now we have to figure
out what to do with the 1

00:15:10.470 --> 00:15:12.530
over x squared.

00:15:12.530 --> 00:15:16.470
But, what do we have here?

00:15:16.470 --> 00:15:17.660
Now I have to remind myself.

00:15:17.660 --> 00:15:20.160
I can't remember all
the steps anymore.

00:15:20.160 --> 00:15:26.710
We have u is equal to x
squared minus y squared.

00:15:26.710 --> 00:15:27.470
Let me come back.

00:15:27.470 --> 00:15:28.970
Now I've forgotten
what I was doing.

00:15:31.600 --> 00:15:32.260
Ah, yes.

00:15:32.260 --> 00:15:34.310
Now I remember, sorry.

00:15:34.310 --> 00:15:34.980
OK.

00:15:34.980 --> 00:15:37.960
So the point I should have
remembered that I forgot,

00:15:37.960 --> 00:15:41.960
is that 1 minus v
squared is equal to u

00:15:41.960 --> 00:15:42.927
divided by x squared.

00:15:42.927 --> 00:15:44.510
That's what I had
figured out earlier,

00:15:44.510 --> 00:15:46.940
that I just forgot when I
was staring at the board.

00:15:46.940 --> 00:15:49.530
And to notice that, what
do we have to remember?

00:15:49.530 --> 00:15:52.730
u is x squared minus y squared,
so if I divide everything

00:15:52.730 --> 00:15:54.220
by x squared, the
first term is 1

00:15:54.220 --> 00:15:55.880
and the second
term is v squared.

00:15:55.880 --> 00:15:57.879
So, whew, that's good.

00:15:57.879 --> 00:15:59.670
So I was a little
nervous there for second,

00:15:59.670 --> 00:16:01.150
but I did in fact
do this earlier.

00:16:01.150 --> 00:16:03.360
And I'd forgotten what I did.

00:16:03.360 --> 00:16:06.930
So now, the 1 minus v
squared is actually the same

00:16:06.930 --> 00:16:09.970
as u divided by x squared.

00:16:09.970 --> 00:16:12.930
And notice what that
does to this term here.

00:16:12.930 --> 00:16:20.770
That tells us that dx*dy over
x squared is actually equal

00:16:20.770 --> 00:16:29.710
to du*dv over-- instead
of the 1 minus v squared,

00:16:29.710 --> 00:16:32.610
I put u over x squared
and I get-- notice,

00:16:32.610 --> 00:16:38.100
I get an x squared times
2, u divided by x squared.

00:16:38.100 --> 00:16:38.600
Right?

00:16:38.600 --> 00:16:42.070
I just replace the 1 minus v
squared with what I know it is,

00:16:42.070 --> 00:16:48.604
the x squareds divide out,
and so I get du*dv over 2u.

00:16:48.604 --> 00:16:50.520
So now the good news is
I have all the pieces,

00:16:50.520 --> 00:16:52.470
because I'm about to
run out of board space.

00:16:52.470 --> 00:16:53.690
So I have all the
pieces, so I'm just

00:16:53.690 --> 00:16:55.800
going to put them together,
and then we're done.

00:16:55.800 --> 00:16:58.620
So let me come here
in the final spot,

00:16:58.620 --> 00:17:02.570
and say this is
our final answer.

00:17:02.570 --> 00:17:08.070
Our final answer is that
we're integrating u from 1,

00:17:08.070 --> 00:17:15.910
to 1 over v minus v. And then
we're integrating v from 0

00:17:15.910 --> 00:17:19.350
to a-- where a is the value
I determined earlier--

00:17:19.350 --> 00:17:26.110
of 1 over 2u, du*dv.

00:17:26.110 --> 00:17:30.330
So this is the
final, final answer.

00:17:30.330 --> 00:17:31.287
This was a long one.

00:17:31.287 --> 00:17:33.620
And I'm sorry I had a little
brain freeze in the middle.

00:17:33.620 --> 00:17:36.060
I couldn't remember how
I'd fixed that problem.

00:17:36.060 --> 00:17:38.140
So what I did at
that point-- I just

00:17:38.140 --> 00:17:41.780
want to point out that when I
was working on this problem,

00:17:41.780 --> 00:17:44.870
and I had a 1 minus v
squared, I knew somehow

00:17:44.870 --> 00:17:47.180
I had to figure out
how to relate that

00:17:47.180 --> 00:17:50.560
and the x squared
in terms of u and v.

00:17:50.560 --> 00:17:53.010
And so I actually
saw this expression.

00:17:53.010 --> 00:17:54.895
I could have written
it better, maybe, as x

00:17:54.895 --> 00:17:56.970
squared times this equals u.

00:17:56.970 --> 00:17:57.470
OK.

00:17:57.470 --> 00:17:58.830
And maybe that would
have been more obvious,

00:17:58.830 --> 00:18:00.020
if that's the case.

00:18:00.020 --> 00:18:02.130
But that was really
the step that

00:18:02.130 --> 00:18:04.740
allowed me to
replace all of this

00:18:04.740 --> 00:18:06.379
by things in terms
of u and v. Which

00:18:06.379 --> 00:18:07.920
I know I should have
been able to do,

00:18:07.920 --> 00:18:09.555
it's just a matter
of figuring it out.

00:18:09.555 --> 00:18:11.180
So let me just go
back to the beginning

00:18:11.180 --> 00:18:13.470
and remind you of each of
the steps very briefly,

00:18:13.470 --> 00:18:15.900
and then we'll be done.

00:18:15.900 --> 00:18:18.510
So we come back over
to the beginning.

00:18:18.510 --> 00:18:22.090
We were starting with change
of variables supplied for us.

00:18:22.090 --> 00:18:24.510
We already had an integral
in terms of x and y,

00:18:24.510 --> 00:18:26.560
and we had an infinite region.

00:18:26.560 --> 00:18:28.260
And what we were
asked to do is find

00:18:28.260 --> 00:18:29.770
the limits and the integrand.

00:18:29.770 --> 00:18:33.170
So the first step
for me is I always

00:18:33.170 --> 00:18:37.980
find it very helpful to draw
the region in the xy-plane,

00:18:37.980 --> 00:18:40.500
and then draw the new
region in the uv-plane.

00:18:40.500 --> 00:18:42.470
Neither one of them
has to be perfect,

00:18:42.470 --> 00:18:48.750
but the understanding of the
values of the curves in terms

00:18:48.750 --> 00:18:53.780
of equations of u and v are very
important, to understand that.

00:18:53.780 --> 00:18:57.370
That gives you the
bounds, the limits.

00:18:57.370 --> 00:18:59.797
And then, so we
did all this work.

00:18:59.797 --> 00:19:00.630
We found the limits.

00:19:00.630 --> 00:19:02.338
There was a little
algebra in the middle.

00:19:02.338 --> 00:19:03.270
We found the limits.

00:19:03.270 --> 00:19:05.780
And then we found
the Jacobian, which

00:19:05.780 --> 00:19:09.640
was going to tell us how
the variables were changing.

00:19:09.640 --> 00:19:11.480
We found it in terms of x and y.

00:19:11.480 --> 00:19:14.520
We rewrote it in
terms of u and v.

00:19:14.520 --> 00:19:19.870
And so when we came back and we
compared what our integrand was

00:19:19.870 --> 00:19:25.424
initially, we could
compare dx*dy to du*dv.

00:19:25.424 --> 00:19:26.840
But then we also
had to figure out

00:19:26.840 --> 00:19:29.170
how to replace the
1 over x squared.

00:19:29.170 --> 00:19:33.550
So once we did all that, we had
everything in terms of u and v,

00:19:33.550 --> 00:19:36.045
and then we finally had what
the integrand was going to be.

00:19:36.045 --> 00:19:38.265
So there were a lot of steps,
but this was ultimately

00:19:38.265 --> 00:19:39.140
what the problem was.

00:19:39.140 --> 00:19:42.830
And again, I'll just point
out, this is the final solution

00:19:42.830 --> 00:19:43.620
right here.

00:19:43.620 --> 00:19:47.780
We integrated from 1, to
1 over v minus v, for u.

00:19:47.780 --> 00:19:52.970
And we integrated from 0 to a
in v, the function 1 over 2u.

00:19:52.970 --> 00:19:53.470
OK.

00:19:53.470 --> 00:19:55.497
That is where I will stop.