## Proof That Polar Moment of inertia Equals Sum of Moments of Inertia Aboput x and y Axes

The polar moment of inertia of a lamina with surface\[S\]

about a axis through the origin perpendicular to the \[xy\]

plane is given by \[I_P \int_S r^2 \rho (x,y) ddS\]

where \[r^2 =x^2 +y^2 \]

The moment of inerta of the lamina about the

\[x\]

and \[y\]

axes are \[I_x = \int_S x^2 \rho (x,y) dS\]

and \[I_y = \int_S y^2 \rho (x,y) dS\]

respectively.Adding these gives

\[I_x +I_y = \int_S x^2 \rho (x,y) dS + \int_S Y^2 \rho (x,y) dS = \int_S (x^2 +y^2 ) dS =I\]