Theorem of perpendicular axes about moment of inertia

Theorem of perpendicular axes about moment of inertia : The moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular axes in its plane and through the point of intersection of the perpendicular axis and the lamina.

Proof: Let Ox and Oy be two perpendicular axes in the plane of the lamina and Oz, an axis perpendicular to its plane. Consider an infinitesimal mass element dm of the lamina at the point P(x, y). MI of the lamina about the z-axis, lz= integration sing op² dm

The element is at perpendicular distance y and x from the x- and y-axes respecti Hence, the moments of inertia of the lamina about the x- and y-axes are, respective