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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 of

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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

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

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over x squared, dxdy.

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And R is the infinite region in
the first quadrant that is

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both under the curve y equals 1
over x, and to the right of

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the curve 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 want to find the limits

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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 you find

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the appropriate limits
and integrand--

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and then when you feel
comfortable, bring the video

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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 is this change of variables

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

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squared, v to be y divided by
x, and we have this region

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that is in the first quadrant
and it's under the curve y

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equals 1 divided by x and it's
to the right of the curve x

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squared minus y squared
equals 1.

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And we want to compute the
limits and integrand for that

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particular integral.

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So what I'm going to do to try
and make this as organized as

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

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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 R

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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 that I

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

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integrating, I will put
those two together to

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figure out the integrand.

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So there are a bunch of steps
to these problems. But the

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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 of

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

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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

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equals 1 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 part of the hyperbola

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that is x squared minus
y squared equals 1.

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So I'll draw in the part
we need, which looks

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roughly like this.

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Something like that.

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

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you an idea of what the
region looks like.

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And the only thing I'm going to
mention is that this point

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we know is x equals
1 and y equals 0.

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So the region we're interested
in that is both to the right

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of x squared minus y squared
equals 1 and below y equals 1

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over x and in the first quadrant
is exactly this

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region I'm shading here.

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

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along these bounds.

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We need to understand where this
region maps to when I do

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the change of variables in order
to understand what the

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limits are.

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

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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 know

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this is going into the
first quadrant.

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So it doesn't always work that
something in the first

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quadrant maps into the first
quadrant, but in this case, I

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already did the work,
so I know it does.

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So let me point out a few
things about where

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this region R maps.

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The first thing I want to point
out is that we actually

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know that this curve under
the change of variables

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maps to u equals 1.

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Because if you remember,
u is equal to x

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squared minus y 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 some part of the

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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
zero, if I look at what v is--

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

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equal 1, y equals 0-- v is 0.

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

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here-- my starting point
on this segment is

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actually also at 1, 0.

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OK?

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So I know there's some point
right here that maps down to

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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 we'll figure out what

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that point is, 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 to rewrite what the

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change of variables is here,
so I don't have to keep

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walking over to the
other side.

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Our change of variables was u is
equal to x squared minus y

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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, and v is

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going to equal 0.

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And so the point of this,
really, is-- that even though

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in u, this curve maybe is
mapping at a different speed

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in some form to this
curve here--

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it still it's just taking that
segment goes, or that

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infinitely long ray goes to an
infinitely long ray here along

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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 that ray and u

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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, because we're

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going to say this is part of
the region, or that's one

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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, but I can still

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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-- is y equals 1

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divided by x.

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So that means that
on that curve--

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let me even write it down-- on
y equals 1 divided by x, v is

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equal to 1 divided by
x divided by x.

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

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And then what does that
mean about u?

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u, then, is equal to--

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well, x squared is 1 divided
by v, and then y squared,

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because y squared on that curve
is just 1 divided by x

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squared, is v. So let me just
make sure we all followed that

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one more time.

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We're looking at where the curve
y equals 1 over x goes

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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. So we

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want to know where that goes.

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

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squared, because v-- we
know-- 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, this means x

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

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becomes 1 over v. And
then y squared--

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which is 1 over x squared--

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become v. So that curve is u
equals 1 over v minus v.

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Now that curve, roughly,
is going to look

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something like this.

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And it might seem strange.

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The thing is, I'm graphing this
in the uv plane, and I'm

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writing what looks like u as a
function of v, and so it's

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sort of turned around from
how you usually see

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these things written.

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But this is the equation that
describes this curve.

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And that is sufficient to
understand, because when we

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use, determine our bounds, we
can determine our bounds from

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u equals 0 now, to u equals 1
over v minus v. So we now have

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the bounds in u.

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We're actually doing
quite well.

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So we have this region.

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We now have the bounds
completely in u.

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u is going from u equals 0 to u
equals 1 over v minus v. But

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the problem is now we don't know
the bounds in v. We don't

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know what the bounds are in v,
and so we have to be a little

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bit careful.

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So actually, no.

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I think I was wrong.

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It's not 0, is it?

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I said that twice now, and
that was incorrect.

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u is going from 1, to 1 over
v minus v. So I apologize.

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Because the slices of u are
going from whatever the

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u-value starts-- which is
at the value 1-- and

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it's coming this way.

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So I apologize.

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I was moving my arm like I was
doing the v-values, but I

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actually wanted to
do the u-values.

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So I want to go from
where u starts--

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which is at u equals 1--

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to where u stops-- which is when
it hits the curve 1 over

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v minus v equals u.

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So hopefully I didn't confuse
anyone by that.

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I'm glad I caught it, then.

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OK, so now we understand
the bounds in u.

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And then to understand the
bounds in v, all we need to

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understand is what is the
v-value at this point.

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So once I know the v-value at
this point, then I'm done with

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the bounds.

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So let's see if we
can find that.

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Well, the v-value at that point
is going to be at the

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point where these two
curves intersect.

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So let's see if we can do a
little algebra to understand

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what that will look like.

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So let me point out that where
those curves intersect, I have

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the equation x squared minus 1
over x squared is equal to 1.

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And if I want to find x-values
that satisfy this, I'm also

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looking for x-values that
satisfy x to the fourth minus

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1 is equal to x squared, which
I can rewrite as x to the

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fourth minus x squared minus
1 is equal to 0.

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So I can actually use the
quadratic formula on this in

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terms of x squared.

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So what I get is I get x
squared is equal to 1--

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I get plus or minus root 5--

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over 2.

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And if you look at it, the one
you're actually interested

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in-- you can figure this out
pretty quickly-- is the one

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that is plus.

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OK?

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I want the one that is
plus root 5 over 2.

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So then that means x is the
square root of this quantity

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at that point, right?

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Or I could actually think
about it this way.

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Let me point out this. v is
equal to 1 over x squared at

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that point, because it's on
that curve where we were

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talking about y equals
1 over x.

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So v is 1 over x squared.

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00:11:46,940 --> 00:11:51,530
So 1 over x squared is just
1 over this quantity.

244
00:11:51,530 --> 00:11:52,950
So it's the reciprocal
of this.

245
00:11:52,950 --> 00:11:57,640
It's also negative 1 plus
root 5, over 2.

246
00:11:57,640 --> 00:11:59,460
You can check that
if you need to.

247
00:11:59,460 --> 00:12:02,520
But I will write it down this
way as the following: this is

248
00:12:02,520 --> 00:12:05,790
the point 1 comma a.

249
00:12:05,790 --> 00:12:11,690
And if I come over here, I
will denote a will equal

250
00:12:11,690 --> 00:12:15,480
negative 1 plus root
5, over 2.

251
00:12:15,480 --> 00:12:17,810
And that's really just 1
divided by x squared.

252
00:12:17,810 --> 00:12:21,880
So let me point that out again,
that a is equal to 1

253
00:12:21,880 --> 00:12:26,195
divided by x squared at the
point of intersection.

254
00:12:26,195 --> 00:12:32,220

255
00:12:32,220 --> 00:12:34,540
So hopefully you can
see all that.

256
00:12:34,540 --> 00:12:38,410
So that tells us our
bounds completely.

257
00:12:38,410 --> 00:12:40,480
We still have some work to do.

258
00:12:40,480 --> 00:12:42,660
So I'm going to put in the
bounds and I'm going to leave

259
00:12:42,660 --> 00:12:44,330
an empty space.

260
00:12:44,330 --> 00:12:44,800
Actually, no.

261
00:12:44,800 --> 00:12:46,820
I won't do that, because
this can get a

262
00:12:46,820 --> 00:12:47,860
little messy and confusing.

263
00:12:47,860 --> 00:12:50,450
So I'm just going to do the
Jacobian, and then we'll

264
00:12:50,450 --> 00:12:52,270
figure it all out and write the
answer right at the end,

265
00:12:52,270 --> 00:12:53,760
so there's no confusion.

266
00:12:53,760 --> 00:12:56,410
But hopefully you see at this
point that we have the bounds.

267
00:12:56,410 --> 00:13:00,580
We know that u goes from 1, to
1 over v minus v. And v goes

268
00:13:00,580 --> 00:13:05,400
from 0 up to a, where a is the
value I've written here.

269
00:13:05,400 --> 00:13:07,210
So we know the bounds.

270
00:13:07,210 --> 00:13:10,250
So now we have to figure
out the integrand.

271
00:13:10,250 --> 00:13:14,350
So let's first compute
the Jacobian, OK?

272
00:13:14,350 --> 00:13:22,200
So now we're looking at del u,
v, del x, y using the notation

273
00:13:22,200 --> 00:13:23,410
we've seen in class.

274
00:13:23,410 --> 00:13:26,840
And so del uv, del xy is going
to be the determinant of the

275
00:13:26,840 --> 00:13:30,690
following matrix:
2x, negative 2y.

276
00:13:30,690 --> 00:13:33,730
And then the derivative respect
to v of x is negative

277
00:13:33,730 --> 00:13:36,340
y over x squared.

278
00:13:36,340 --> 00:13:39,810
And the derivative of v with
respect to y is just 1 over x.

279
00:13:39,810 --> 00:13:49,850
So if I take the determinant of
that, I get 2, minus 2, y

280
00:13:49,850 --> 00:13:52,600
squared over x squared.

281
00:13:52,600 --> 00:13:55,600
Which if you notice, in terms of
our change of variables, is

282
00:13:55,600 --> 00:14:01,630
exactly equal to 2
minus 2v squared,

283
00:14:01,630 --> 00:14:04,370
because v is y over x.

284
00:14:04,370 --> 00:14:06,410
And so I can rewrite
this as 2 times the

285
00:14:06,410 --> 00:14:10,010
quantity 1 minus v squared.

286
00:14:10,010 --> 00:14:11,190
OK?

287
00:14:11,190 --> 00:14:13,910
So, so far so good, hopefully.

288
00:14:13,910 --> 00:14:18,900
Now let's figure out how
to do the final step.

289
00:14:18,900 --> 00:14:20,820
So the final step--

290
00:14:20,820 --> 00:14:22,750
I'm going to come back over and
just remind us what the

291
00:14:22,750 --> 00:14:24,795
integrand was, OK?

292
00:14:24,795 --> 00:14:28,230
If we come over here, we're
reminded that we were

293
00:14:28,230 --> 00:14:33,360
integrating over the region R
of 1 over x squared, dxdy.

294
00:14:33,360 --> 00:14:33,550
Right?

295
00:14:33,550 --> 00:14:35,760
That's what we were interested
in initially.

296
00:14:35,760 --> 00:14:40,890
So now, if we come back, I'm
going to write that down just

297
00:14:40,890 --> 00:14:42,140
to have it as a reference.

298
00:14:42,140 --> 00:14:48,480

299
00:14:48,480 --> 00:14:50,130
OK, that's what we
had initially.

300
00:14:50,130 --> 00:14:51,220
Let me make sure.

301
00:14:51,220 --> 00:14:52,950
Yes, that's what we
had initially.

302
00:14:52,950 --> 00:15:03,030
And so now we know dxdy is equal
to dudv over 2 times 1

303
00:15:03,030 --> 00:15:04,620
minus v squared.

304
00:15:04,620 --> 00:15:07,730
So that is going to
replace the dxdy.

305
00:15:07,730 --> 00:15:10,170
And now we have to figure
out what to do with

306
00:15:10,170 --> 00:15:12,530
the 1 over x squared.

307
00:15:12,530 --> 00:15:16,470
But, what do we have here?

308
00:15:16,470 --> 00:15:17,660
Now I have to remind myself.

309
00:15:17,660 --> 00:15:20,160
I can't remember all
the steps anymore.

310
00:15:20,160 --> 00:15:26,710
We have u is equal to x squared
minus y squared.

311
00:15:26,710 --> 00:15:27,470
Let me come back.

312
00:15:27,470 --> 00:15:28,720
Now I've forgotten
what I was doing.

313
00:15:28,720 --> 00:15:31,600

314
00:15:31,600 --> 00:15:32,260
Ah, yes.

315
00:15:32,260 --> 00:15:34,310
Now I remember, sorry.

316
00:15:34,310 --> 00:15:34,980
OK.

317
00:15:34,980 --> 00:15:38,230
So the point I should have
remembered that I forgot, is

318
00:15:38,230 --> 00:15:41,960
that 1 minus v squared
is equal to u

319
00:15:41,960 --> 00:15:43,210
divided by x squared.

320
00:15:43,210 --> 00:15:45,360
That's what I had figured out
earlier that I just forgot

321
00:15:45,360 --> 00:15:46,940
when I was staring
at the board.

322
00:15:46,940 --> 00:15:49,530
And to notice that, what
do we have to remember?

323
00:15:49,530 --> 00:15:52,420
u is x squared minus y squared,
so if I divide

324
00:15:52,420 --> 00:15:54,640
everything by x squared, the
first term is 1 and the second

325
00:15:54,640 --> 00:15:55,880
term is v squared.

326
00:15:55,880 --> 00:15:58,030
So, whew, that's good.

327
00:15:58,030 --> 00:16:00,160
So I was a little nervous there
for second, but I did in

328
00:16:00,160 --> 00:16:01,150
fact do this earlier.

329
00:16:01,150 --> 00:16:03,360
And I'd forgotten what I did.

330
00:16:03,360 --> 00:16:07,960
So now, the 1 minus v squared
is actually the same as u

331
00:16:07,960 --> 00:16:09,970
divided by x squared.

332
00:16:09,970 --> 00:16:12,930
And notice what that does
to this term here.

333
00:16:12,930 --> 00:16:21,850
That tells us that dxdy over x
squared is actually equal to

334
00:16:21,850 --> 00:16:24,313
dudv over--

335
00:16:24,313 --> 00:16:26,870

336
00:16:26,870 --> 00:16:30,930
instead of the 1 minus v
squared, I put u over x

337
00:16:30,930 --> 00:16:32,330
squared and I get--

338
00:16:32,330 --> 00:16:36,110
notice, I get-- an x
squared times 2, u

339
00:16:36,110 --> 00:16:38,300
divided by x squared.

340
00:16:38,300 --> 00:16:38,600
Right?

341
00:16:38,600 --> 00:16:41,440
I just replace the 1 minus v
squared with what I know it

342
00:16:41,440 --> 00:16:48,700
is, the x squareds divide out,
and so I get dudv over 2u.

343
00:16:48,700 --> 00:16:50,750
So now the good news is I have
all the pieces, because I'm

344
00:16:50,750 --> 00:16:52,470
about to run out
of board space.

345
00:16:52,470 --> 00:16:54,200
So I have all the pieces, so
I'm just going to put them

346
00:16:54,200 --> 00:16:55,800
together, and then we're done.

347
00:16:55,800 --> 00:17:00,000
So let me come here in the final
spot, and say this is

348
00:17:00,000 --> 00:17:02,570
our final answer.

349
00:17:02,570 --> 00:17:09,540
Our final answer is that we're
integrating u from 1, to 1

350
00:17:09,540 --> 00:17:16,920
over v minus v. And then we're
integrating v from 0 to a--

351
00:17:16,920 --> 00:17:19,350
where a is the value I
determined earlier--

352
00:17:19,350 --> 00:17:30,330
of 1 over 2u, dudv. So this is
the final, final answer.

353
00:17:30,330 --> 00:17:31,700
This was a long one.

354
00:17:31,700 --> 00:17:33,620
And I'm sorry I had a little
brain freeze in the middle.

355
00:17:33,620 --> 00:17:36,060
I couldn't remember how I'd
fixed that problem.

356
00:17:36,060 --> 00:17:37,860
So what I did at that point--

357
00:17:37,860 --> 00:17:40,700
I just want to point out-- that
when I was working on

358
00:17:40,700 --> 00:17:44,590
this problem, and I had a 1
minus v squared, I knew

359
00:17:44,590 --> 00:17:47,970
somehow I had to figure out how
to relate that and the x

360
00:17:47,970 --> 00:17:50,560
squared in terms of u and v.

361
00:17:50,560 --> 00:17:53,010
And so I actually saw
this expression.

362
00:17:53,010 --> 00:17:55,150
I could have written it better,
maybe, as x squared

363
00:17:55,150 --> 00:17:57,120
times this equals u.

364
00:17:57,120 --> 00:17:57,470
OK.

365
00:17:57,470 --> 00:17:58,830
And maybe that would have
been more obvious,

366
00:17:58,830 --> 00:18:00,020
if that's the case.

367
00:18:00,020 --> 00:18:04,160
But that was really the step
that allowed me to replace all

368
00:18:04,160 --> 00:18:07,235
of this by things in terms of u
and v. Which I know I should

369
00:18:07,235 --> 00:18:08,490
have been able to do,
it's just a matter

370
00:18:08,490 --> 00:18:09,700
of figuring it out.

371
00:18:09,700 --> 00:18:11,940
So let me just go back to the
beginning and remind you of

372
00:18:11,940 --> 00:18:13,630
each of the steps very
briefly, and

373
00:18:13,630 --> 00:18:15,900
then we'll be done.

374
00:18:15,900 --> 00:18:18,510
So we come back over
to the beginning.

375
00:18:18,510 --> 00:18:22,090
We were starting with change of
variables supplied for us.

376
00:18:22,090 --> 00:18:24,970
We already had an integral in
terms of x and y, and we had

377
00:18:24,970 --> 00:18:26,560
an infinite region.

378
00:18:26,560 --> 00:18:28,320
And what we were asked
to do is find the

379
00:18:28,320 --> 00:18:29,770
limits and the integrand.

380
00:18:29,770 --> 00:18:35,500
So the first step for me is I
always find it very helpful to

381
00:18:35,500 --> 00:18:38,810
draw the region in the xy plane,
and then draw the new

382
00:18:38,810 --> 00:18:40,500
region in the uv plane.

383
00:18:40,500 --> 00:18:44,770
Neither one of them has to be
perfect, but the understanding

384
00:18:44,770 --> 00:18:51,230
of the values of the curves in
terms of equations of u and v

385
00:18:51,230 --> 00:18:53,780
are very important, to
understand that.

386
00:18:53,780 --> 00:18:57,370
That gives you the bounds,
the limits.

387
00:18:57,370 --> 00:18:59,880
And then, so we did
all this work.

388
00:18:59,880 --> 00:19:00,630
We found the limits.

389
00:19:00,630 --> 00:19:01,710
There was a little algebra
in the middle.

390
00:19:01,710 --> 00:19:03,270
We found the limits.

391
00:19:03,270 --> 00:19:06,460
And then we found the Jacobian,
which was going to

392
00:19:06,460 --> 00:19:09,640
tell us how the variables
were changing.

393
00:19:09,640 --> 00:19:11,480
We found it in terms
of x and y.

394
00:19:11,480 --> 00:19:14,520
We rewrote it in terms
of u and v.

395
00:19:14,520 --> 00:19:19,680
And so when we came back and we
compared what our integrand

396
00:19:19,680 --> 00:19:25,880
was initially, we could compare
dxdy to dudv. But then

397
00:19:25,880 --> 00:19:28,070
we also had to figure
out how to replace

398
00:19:28,070 --> 00:19:29,170
the 1 over x squared.

399
00:19:29,170 --> 00:19:32,500
So once we did all that, we had
everything in terms of u

400
00:19:32,500 --> 00:19:35,115
and v, and then we finally
had what the integrand

401
00:19:35,115 --> 00:19:36,045
was going to be.

402
00:19:36,045 --> 00:19:38,730
So there were a lot of steps,
but this was ultimately what

403
00:19:38,730 --> 00:19:39,140
the problem was.

404
00:19:39,140 --> 00:19:42,400
And again, I'll just point
out, this is the final

405
00:19:42,400 --> 00:19:43,620
solution right here.

406
00:19:43,620 --> 00:19:47,780
We integrated from 1, to 1
over v minus v, for u.

407
00:19:47,780 --> 00:19:53,040
And we integrated from 0 to a in
v, the function 1 over 2u.

408
00:19:53,040 --> 00:19:53,470
OK.

409
00:19:53,470 --> 00:19:55,550
That is where I will stop.

410
00:19:55,550 --> 00:19:55,996