In converting to spherical coordinates, we use
[tex]x=\rho\cos\theta\sin\varphi[/tex]
[tex]y=\rho\sin\theta\sin\varphi[/tex]
[tex]z=\rho\cos\varphi[/tex]
so that
[tex]\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
Then the integral is
[tex]\displaystyle\iiint_Ey^2\,\mathrm dV=\int_0^\pi\int_0^\pi\int_0^3\rho^4\sin^2\theta\sin^3\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta=\boxed{\frac{162\pi}5}[/tex]
