explain moment of inertia of thin rod , solid sphere, ring

Here are the derivations of MI of rod, ring, cylinder and sphere.

Moment of inertia of a cylinder (hollow) of radius r about its central axis (shown in red line)

Moment of inertia of a cylinder (solid) of radius r about its central axis (shown in red line) 

Before we derive the MI of solid cylinder let us derive the MI of a circular disk.

Consider a small area strip element at a distance x from the centre of the disk of radius R.  Let mass of the disk = m