But setting up the integral is the only interesting part of this problem!

Since this involves anupside downcone, it's probably best to use cylindrical coordinates (if it had been a "regular" cone, spherical coordinates). In cylindrical coordinates the sphere is given by, of course, and the cone is given by z= 4- r. The differential of volume is . The cone and the sphere intersect when or . That is, they intersect at r= 0 (the top of the sphere) and at r= 4, the circumference.

To cover the entire sphere, r must go from 0 to 2 and must go from 0 to . For each r, z must go from the cone z= 4- r up to the upper half of the sphere, z= \sqrt{16- r^2}.