Chapter 15.8, Problem 42E

Evaluate the integral by changing to spherical coordinates.

42. ${\displaystyle {\int }_{-a}^a}{\displaystyle {\int }_{-\sqrt{a^2-y^2}}^{\sqrt{a^2-y^2}}}{\displaystyle {\int }_{-\sqrt{a^2-x^2-y^2}}^{\sqrt{a^2-x^2-y^2}}}$ (x²z + y²z + z³) dz dx dy