By definition of polar co-ordinates our integral transforms from
So now we need bounds for r, theta and z.
In the case of this problem our Z is bounded by 2 functions. Let us label them Z1 and Z2. So to get the area between these 2 functions we need to subtract the smaller Z from the bigger Z. What this is in effect, is the integral of
But now we need our bounds for R. Well, if we put out equations together, this yields
Which is a circle of radius 1. What this means is, we are integrating over this domain in the XY plane. Notice that
Therefore, r runs from the origin (0) to 1.
Now we need bounds for our theta...But remember, this is a circle! So how many degrees are there in a circle? That would be 360 or
So you see, what we are essentially doing is integrating from 0-->360 over a circle with radius R projected upward into the Z plane.
Does this make sense?