Question

A cylindrical drill with radius 4 is used to bore a hole through the center of a sphere of radius 5. Find the volume of the ring shaped solid that remains.

Answer 1

This is actually easy. You want the region outside of the cylinder of radius 3 and inside a sphere of radius 5. This means that you want:
x^2 + y^2 > 9 and x^2 + y^2 + z^2 < 25
To integrate it is easiest to first switch to cylindrical coordinates. The Jacobian determinant in cylindrical coordinates is "r". You want to integrate rdtheta dz dr over the limits:
3<r<5, -sqrt[25-r^2]<z<sqrt[25-r^2], 0<theta<2pi
The integration is easy to do in this form and you get an exact answer of 256pi/3. This would be about 268.083 cubic units of volume.