00:01
In this problem, we want to evaluate the given integral by changing the polar coordinates.
00:06
So here we have a double integral of 9 times x plus y da over region r, where r is the region on the left of the y -axis between the circles of radius 1 and radius 4.
00:20
So let's start by drawing the area, the region for our area integral.
00:31
We want to integrate between the circles of area 1 and a circle of radius 4.
00:46
In addition, we're only interested in the area on the left of the y -axis, which corresponds to this region right here.
00:58
So this here is our region r.
01:07
Now let's recall how do we change from cartesian coordinates xy to polar coordinates r theta.
01:14
We have that x is defined as r times cos theta, and y is defined as r times sin theta.
01:25
In addition, the area element, which traditionally in cartesian coordinates is given by the product of dx times dy, will now be written as r dr d theta.
01:42
In case you're wondering where that extra r comes from, this just arises from the jacobian.
01:48
You can then verify your value on your own time, which arises when you change variables in multivariable calculus.
02:00
So now we have almost everything we need to rewrite our integral in polar coordinates.
02:06
So we will be integrating 9 times r times cos theta plus sin theta.
02:17
This is x plus y times r dr d theta...