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?