Calculate a triple integral where the integrand contains cubic terms

Question

Calculate $$I = \iiint\limits_Q(x^3 + y^3 + z^3)\,dx\,dy\,dz$$ where $Q: x^2 + y^2 + z^2 - 2ax-2ay-2az + 2a^2 = 0$ is a sphere

If I try using spherical coordinates I don't go anywhere. Any hint about how to calculate it?

Answer 1

Due to symmetry $$I = \iiint_Q(x^3 + y^3 + z^3)\,dV =3 \iiint_Q z^3\,dV$$ Then, recenter the sphere at origin and integrate in spherical coordinates \begin{align} I =& \ 3 \iiint_{x^2+y^2+z^2<a^2} (z+a)^3\,dV =\ 3 \iiint_{x^2+y^2+z^2<a^2} (3az^2 +a^3)\,dV\\ =& \ 3a \iiint_{r<a} r^2 \ r^2\sin\phi dr d\theta d\phi + 3a^3 \cdot \frac43\pi a^3=\frac{32}5\pi a^6 \end{align}