DAVID JORDAN: Hello, and welcome back to recitation.
So today I want to practice doing some computings of
integrals with you at polar coordinates.
So we have these three integrals set up here.
And the way the integrals are given to you, they're given to
you in rectangular coordinates.
So the first thing you need to do is to re-express these in
polar coordinates.
And so for the first one, part a, I want you to completely
compute the integral.
For part b and c, we have this function f, which we haven't
specified yet.
So the exercise is to set up the integral.
We won't actually compute it, we'll just set it up
completely.
So rewrite it in terms of r and theta.
So why don't you pause the video and get started on that.
And check back with me and we'll work it out together.
Welcome back.
Let's get started.
So for all of these, when we're transferring from
rectangular to polar coordinates, the most
difficult part is understanding what region
we're integrating over.
And if we can understand that, then the rest of it is just
straightforward calculation.
So in part a, let's try to think about what this region
is that we're given.
So we're given the region from x equals 1 to x equals 2.
So let's draw those lines. x equals 1 and x equals 2.
OK, so that's how x varies in between this band.
And now the y varies from 0 to the function x.
So we'd better draw the line y equals x.
OK, so this is the line y equals x.
And the region that we're after is in between this band
and in between these two lines, so it's this region
right here.
That's the region that we want.
All right.
So now that we have that, what we need to do to express
something in polar coordinates is we need to sweep out rays
of constant angle, and then measure the
radius along the ray.
So let's see what I mean by that.
So we can kind of see that if we're inside this region here,
the smallest value that theta can be is just theta equals 0.
Because, sprint is right here is a point at theta equals 0.
And then theta runs all the way up.
And this whole line here is where theta reaches its
maximum, and that's at theta is pi over 4.
So this tells us that it's easiest to write the theta
integral on the outside.
And we're going to have theta running from 0 to pi over 4.
OK, very good.
Now, in order to get the r ranges, it's going to be a
little bit more subtle, because what we need to do is
we need to fix some arbitrary theta in the middle here.
So I'll just draw kind of a representative theta.
It would be like this one here.
So there is a representative theta.
And what we need to do is, we need to describe how r varies
along this pink line here.
So that's the line that we really want.
OK.
So in order to do that, we just need to do some simple
trigonometry now.
So let's see.
So r turns on at this line right here.
At the line x equals 1: that's where r turns on.
So let's see what the value of r is at that line here.
So, the pont is that we have --so let me draw
this triangle here--.
Let me blow this triangle up.
OK.
So this is our line x equals 1, and this is
this triangle here.
We just put a magnifying glass on it.
So we have this angle theta.
OK.
So this x-value is 1, because we're on the line x equals 1.
So the length of this leg is 1, and this is
r that we're after.
OK?
So in basic trigonometry we know that cos
theta is equal to--
so cosine is adjacent over hypotenuse--
is equal to 1 over r.
OK, so that tells us that r is equal to sec theta.
OK?
So what that means is that this, for any given theta,
this pink band turns on at precisely r equals sec theta.
And so that's what we need to write in the bottom here.
OK?
Now, you can see that it turns off at r equals 2 sec theta.
Because now instead of going over length 1, we
go over length 2.
And so our integral becomes the integral from
0 to pi over 4.
And r equals sec theta to 2 sec theta.
OK, and now we need to re-express our original
function in terms of r and theta.
So our original function was, so let's go back over to the
formula here.
So we have x squared plus y squared to the 3/2.
So x squared plus y squared to the 1/2 that's r.
And so x squared plus y squared to the
3/2 that's r cubed.
So this is 1 over r cubed.
So this is 1 over r cubed in here.
And as we know, dy dx becomes r dr d theta.
OK.
So this is the integral that we need to compute, and now
that we've done all the geometry, this is going to be
a straightforward integral to compute.
So let's do it together.
I'll just rewrite it over here.
So we have theta running from 0 to pi over 4, and r running
from sec theta to 2 sec theta.
And we have 1 over r squared, dr d theta.
OK.
So that equals--
taking the inside integral first--
we get negative 1 over r from 2 sec theta to sec theta.
OK?
And now this is nice, because sec theta--
if we plug it in for 1 over r-- it's just going to turn
back into a cosine, right?
So this is just the integral from theta equals
0 to pi over 4.
OK, because we have this minus sign here, we're going to get
cos theta minus--
1 over 2 sec theta becomes--
1/2 cos theta, d theta.
And this we can compute to be the square root of 2 over 4.
So this is an elementary integral that we can compute:
the square root of 2, over 4.
OK.
And that's it.
That's all there is to a.
So notice that the difficult part is figuring out how the
boundary curves of your original region re-express in
the r and theta coordinates.
So let's see if we can get more practice with that on
parts b and c.
So in part b--
let me just recall--
we're taking the integral x equals 0 to 1.
And y equals x squared to x, f dy dx.
So let's draw this region again.
So the bottom curve is y equals x squared and the top
curve is y equals x, and x runs from 0 to 1.
So this is the region that we're after.
OK.
So once again in this case, it's pretty easy to figure out
the range for theta.
Because you see this parabola here, as it approaches the
origin, it becomes flat.
So the theta, as we get closer and closer to
the origin, is 0.
So the initial bound for theta is 0.
It doesn't matter that we have this curve here, it's still 0.
And then the top bound, again, is pi over 4.
OK?
Now, for this curve y equals x squared, what we just have to
do is we just have to use our polar change of coordinates
and re-express this curve.
So y equals x squared.
Well, y is r sine theta.
And x is r cos theta.
So this is altogether r squared cos squared theta.
OK?
And now all we need to do is just solve for r here.
So we get r equals-- so I can cancel that r with that one--
and it looks to me like we get r equals tan theta secant
theta by solving.
OK.
So that's the r-value at each of these
points along the curve.
So that is going to be our top bound.
So if we draw little segments of constant theta, they all
start at r equals 0, because we have this sharp point here
at the origin.
And they all stop at this curve: r equals
tan theta sec theta.
All right?
And so now f we're just going to leave put.
And then finally we have r dr d theta.
OK.
So that's our answer to b.
All right.
Now c is going to be a little bit tricky.
What's tricky about c is drawing a picture, but we'll
see what we can do about that.
So let me just remind you. y ranges from 0 to 2, and x
ranges from 0 to the square root of 2y minus y squared.
And then we have f dx dy.
OK.
Well, the y bounds are pretty straightforward, 0 to 2.
This function, when I first saw this, I didn't recognize
this function right away.
So let's see what we can make out of this curve.
So the top curve for x is going to be x equals the
square root of 2y minus y squared.
And this looks like it wants to be the
equation of a circle.
So let's see if we can turn this into an
equation of a circle.
So if we square both sides of this equation, then we get x
squared equals 2y minus y squared.
And now this over here, 2y minus y squared, so if we add
this over to the other side, we have x squared plus y
squared minus 2y equals 0.
And now the thing that I notice about this expression
here is it really looks almost like the
quantity y minus 1 squared.
Yeah?
If there were a plus 1 here, then that would be the case.
So this is actually x squared, plus y minus 1 squared--
except the left-hand side is not quite equal to that-- it's
equal to that, minus 1.
So this equation just equals 0.
So this equation is just equivalent to that one.
And that's good news, because this says that, this is the
equation if I add this 1 over to the other side--
let me go over here--
we have x squared, plus y minus 1 squared, equals 1.
And this, we know what this is.
This is a circle which is centered not at the origin,
but at the point 0 comma 1 in the y-direction.
So if we draw this, well, this is the equation
for the entire circle.
But we only want the positive half, because we were told
that x starts at 0 and goes up to this positive number.
So we only want the positive half here.
OK.
So this is the region that we're integrating.
And now that we know this, we can again figure out the
values of theta and the values of r.
So first of all, theta is going to range again from 0,
because this bottom curve, as it comes into the origin, it
comes in flat.
So we have points of arbitrarily small angles.
So theta is going to start a 0, and theta is going to go
all the way up until pi over 2, because theta can be
pointing straight up.
OK?
And now we have to think about these lines of constant angle,
and we need to think about what r does in there.
So we can see from the picture that r always starts at 0 and
it stops at this curve.
So we need to figure out what the r-value is along this
curve in terms of theta.
So what we want to do is use the equation.
So we had it over here.
So x squared plus y squared minus 2y equals 0.
So that was one of the equivalent equations that we
got for our curve.
And now this is nice, because this here is just r
squared, isn't it?
So this is r squared.
And this is minus 2y, and y is r sine theta.
OK.
So we get r squared minus 2r sine theta equals 0.
Solving this for r, we get that r is just 2 sine theta.
OK.
So r equals 2 sine theta.
That is the equation in r, theta coordinates, which
describes this semicircle here.
And so altogether, we get the range is from r equals 0 to r
equals 2 sine theta.
And f just comes along for the ride, and
we have r dr d theta.
And I'll leave it at that.