Calculating Moment of Inertia for Cylinder and Sphere Intersection

[kasse]

Find the moment of inertia around the x-axis of the region that lies inside both the cylinder r=a and the sphere rho=2a.

*Here's what I've done so far: http://www.badongo.com/pic/574867

Am I on the right track? If so, what's the next step? If not, what's wrong?

 

[quasar987]

Well one mistake is that rho does not go from 0 to 2a. The cylinder is only of radius a.

In any case, what I suggest is to break down the intergral into two parts. In the first, you'll integrate using cylindrical coordinates over the part of the cylinder enclosed in the sphere, and in the second, you'll integrate using spherical coordinates over the two little spherical caps of the cylinder. Be careful with the lower bound of the rho integral while integrating the spherical caps.