$\displaystyle \int_{-2}^{2} \int_{0}^{\sqrt{4-y^{2}}} \int_{-\sqrt{4-x^{2}-y^{2}}}^{\sqrt{4-x^{2}-y^{2}}} y^{2} \sqrt{x^{2}+y^{2}+z^{2}} \ dz \ dx \ dy $

I got it down to $\displaystyle \int_{0}^{\pi} \int_{0}^{2 \pi} \int_{0}^{2} \rho^{5} \sin^{3}\phi \ \sin^{2} \theta \ d \rho \ d \theta \ d \phi $

And from here just separate it out and integrate.

Is this correct?