I tried to do this problem last night but ended up getting swamped by calculations. One way to find the answer is to switch to cylindrical coordinates:

so that the equations for the paraboloids become

Solving for the intersection of the two gives us

for the radius of a circle in the -plane that bounds the solid. Our center of mass for the -coordinate becomes

where is the volume of the solid.

Hint: The - and -coordinates of the center of mass must be zero. Why is this?