(879,#29) Use cylindrical or spherical coordinates, whichever seems appropriate, to find the volume and centroid of the solid E that lies above the cone z = sqrt(x² + y²)

and below the sphere x² + y² + z² = 1.

Solution: We have

The centroid equations are:

From the symmetry of the region and the constant density, M_xz = 0 , M_yz = 0.

Therefore, the centroid of the region is (0,0,3/(16-8sqrt(2)).

by:sh