moment of inertia

The moment of inertia (also called the second moment) of a particle of mass  about an axis is defined to be  , where  is the distance from the particle to the axis. We extend this concept to a lamina with density function  and occupying a region  by proceeding as we did for ordinary moments. We divide  into small rectangles, approximate the moment of inertia of each subrectangle about the, and take the limit of the sum as the number of subrectangles becomes large. The result is the moment of inertia of the lamina about the :

Similarly, the moment of inertia about the  is

Note that 

EXAMPLE 3: Find the moments of inertia  , and  of a homogeneous disk  with density  , center the origin, and radius .

The boundary of  is the circle  and in polar coordinates  is described by . Let’s compute  first:

Instead of computing  and  directly, we use the facts that  and  (from the symmetry of the problem). Thus