Find the volume of the solid that is under the hemisphere $z=\sqrt{1-x^2-y^2}$ above the region bounded by the graph of the circle $x^2 + y^2-y=0$.
I solved this problem using limits of integration $0$ to $\pi/2$. The graph and double integral are found in this image:
This was done using symmetry, yet if I use $0$ to $\pi$ as limits of integration (see this image below), then I do not get the right answer, what goes wrong ? How can it be done properly without using symmetry?
Note: the right answer is approximately $0.60$