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

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