Use spherical coordinates.
Let H be a solid hemisphere of radius 2 whose density at any point is proportional to its distance from the center of the base. (Let K be the constant of proportionality.)
(a) Find the mass of H.
(b) Find the center of mass of H. (Assume the upper hemisphere of a sphere centered at the origin.)
(c) Find the moment of inertia of H about its axis.