Section 15.8: Triple Integration in Spherical Coordinates

Evaluate the integral by changing to spherical coordinates. \[ \int_{-a}^a \int_{-\sqrt{a^2-y^2}}^{\sqrt{a^2-y^2}} \int_{-\sqrt{a^2-x^2-y^2}}^{\sqrt{a^2-x^2-y^2}} (x^2z + y^2z + z^3) \, dz \, dx \, dy \]