## Finding the Moment of Inertia of a Rigid Body by Integration

If we rotate a body about an axis, we will be increasing its angular momentum. The moment of inertia I of a body is a measure of its resistance to increasing angular momentum. If the body is just a single point particle of mass m, a distance r from the axis then

In general a body is extended in space and has a continuous mass distribution. We must use integration to find the moment of inertia using the formula

For example to find the moment of inertia of the uniform lamina illustrated below.

About theaxis the moment of inertia is

by breaking the area into vertical strips and the limits of integration are then

About theaxis the moment of inertia is

as before by breaking the area into vertical strips and the limits of integration are still