how do we derive the moment of inertia of rod, ring, disc, cylinder, sphere and cone?

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