This is certainly not my favorite calculus topic.
I am to solve this by converting to spherical coordinates.
I guessed the region of integration was the volume of the sphere .
So I had:
Went about solving, and..ended up with 0.
NO, that's not correct. You need symmetry in BOTH the region and the integrand
This isn't volume, that's the integral of ONE.
hence
BUT your integrand isn't strictly positive can be both positive and negative and it's balanced in the sense that the positive and negative parts of z cancel each other off to give you the answer of ZERO.