TITLE: $\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$
QUESTION [6 upvotes]: $$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$$
where V is the unit sphere.
No information is given about n but I assume it is an integral. All I could think to do was to convert to spherical co-ordinates and use reduction formulae, but I ended up with really messy answers. Any help would be brilliant.

REPLY [2 votes]: Following runaround's excellent hint, the required integral is 
$$3\int_{-1}^1 z^{2n}\hskip-1.4cm \underbrace{\int\int dx\,dy}_{\mbox{area of disk of radius }\sqrt{1-z^2}} \hskip-1.3cm dz=3\pi \int_{-1}^1 z^{2n}(1-z^2)\,dz={12\pi\over(2n+3)(2n+1)}.$$