First we look at the limits of integration and try to draw the region
we are integrating over. It turns out to be a circle of radius 5 centered
at the origin. You can see this by looking at the limits of the first integral
that you do. The lower limit is -(25-y2)1/2, so the
left boundary of the region has equation x=-(25-y2)1/2.
If we square both sides, this says that 25=y2+x2 with
x non-positive.
Working with the upper limit shows that the points on the right boundary
satisfy 25=y2+x2 with x non-negative. So the whole
boundary is now the circle of radius 5 centered at the origin.
The next step is to find the limits for the integral over the disk in
polar coordinates. Here is a picture of the region of integration.
When we integrated in rectangular coordinates, we drew horizontal lines
to find the limits when we integrated with respect to x, and vertical lines
to get the limits when we integrated with respect to y. If the angle is
constant and r varies, then we get a ray; if r is constant and the angle
changes then we get a circle. So we use circles centered at the origin to
find the limits of the integration with respect to the angle and rays to
find the integral with respect to r.
If we look at our region, we see that our region is a disk, so any circle
centered at the origin which touches our region lies completely inside the
disk, so the lower limit for
the angle integral is 0 and the upper limit is
.
Now the first circle in our disk has a radius of 0, while the last circle
has a radius of 5, so the upper and lower limits of the r integral are 0
and 5. When we put our integral into polar coordinates we get:
When we integrate with respect to the angle, we get:
