(a) Find the principal moments of inertia for a thin disk of mass \(\Delta m\) and radius \(R\), if its mass density is uniform, the origin of coordinates is at the center of the disk, the \(x\) and \(y\) axes are in the plane of the disk, and the \(z\) axis is perpendicular to the disk.

(b) Use this result to help find the principal moments of inertia of a uniform-density sphere of mass \(M\) and radius \(R_{0}\), with origin at the center of the sphere.

(c) The moment of inertia for rotation about the symmetry axis of a ring of mass \(\Delta m\) and radius \(r\) is \(I=\Delta m r^{2}\). Use this fact to help find the moment of inertia about a symmetry axis for a thin spherical shell of mass \(\Delta M\) and radius \(R\), with origin at the center of the shell.

(d) Use the result of part (c) to find the principal moments of inertia of a solid, uniform-density sphere of mass \(M\) and radius \(R_{0}\). Compare with the result of part (b).