Calculation of the moment of inertia of a body

We give here a method for finding the moment of inertia of a uniform thin rod about the centre by calculation.

Consider a rod of mass m and cross-sectional area A, and let r be the mass of the rod per unit volume. Let the rod be divided into slices perpendicular to XY.
(See Figure 1).

For one elementary slice the moment of inertia about 0 is Ardx.x²

For the whole rod:


Similar proofs may be carried out for other simply shaped bodies.