Problem: Find $I_z$ the moment of inertia about the $z-$ axis, of the lamina that covers the square in the plane with vertices $(-1,-1),$ $(1,-1), (1,1),$ and $(-1,1)$, if the density is $\rho (x,y)=x^2y^2.$
Now I know that $$I_z=\int\int\int(y^2+z^2)\rho(x,y,z)dV$$, but with the given information I am unable to set the limits over all integrals. In particular, how do I find a bound on $z$?
PS. I have attached an image of the problem for reference also.
The domain is two dimensional and it lays in the $xy$-plane, therefore use $$I_z=\iint_D(x^2+y^2)\rho(x,y)dxdy.$$ The final result should be $8/15$.