I don't understand how the limits in the solution of part (b) have been obtained. Please explain how it's done.
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.