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