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