Well, the limits from the first integral are given by the line and the section of the circle equation given by .
First, we can see that the first (inner) most integral is with respect to ; thus, we must solve our two equations for . Also, notice the second integral is with respect to . Therefore, we must make sure we only have y's in our second integrand. Solving our two equations for will give us that.
So, solving for we get and .
Therefore, we have as our inside integral. The limits are ordered the way they are, because the circle is above the line.
The outside integral just tells us how far up the y-axis to integrate. So, that’s where the limits are coming from.
And, just a note, when you are dealing with iterated integrals, always make sure your outer most limits are numbers and not functions.