00:01
We are given the integral of a function over a region r, and we are asked to evaluate this integral using polar coordinates.
00:11
The integral is the integral over r of sine of x squared plus y squared a da, where the region r lies in the first quadrant between circles that are centered at zero with radii of 1 and 3.
00:34
So in cartesian coordinates, the region r is going to look something like this.
00:50
We have x and y axes, our origin, and we see that the largest radius is going to be three.
01:10
Our region lies in the first quadrant, and it's between circles in red that are centered at zero.
01:22
So this is our region r.
01:24
In order to write this in polar coordinates, notice that we have two circles, r equals 1 and r equals 3.
01:44
And because we're in the first quadrant, theta is going to range from 0 to pi over 2, so that our region r and polar coordinates is given by, set of all r's and thetas such that r lies between 1 and 3, and theta lies between 0 and pi over 2.
02:16
Since this is a integral region and our function, sine of x squared plus y squared is continuous.
02:27
And this is because x squared is continuous, y square is continuous, so x squared y squared is continuous, and sign is continuous...