- #1

Draconian28

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## Homework Statement

Find [tex] \iiint (x^{2n} + y^{2n} + z^{2n})\,dV [/tex] where the integral is taken over the region of 3D space where [tex] x^{2} + y^{2} + z^{2} \leq 1 [/tex]

## Homework Equations

## The Attempt at a Solution

I tried doing this in Cartesian coordinates, but the limits of integration got very messy and I got stuck after doing the first integral. I also tried using spherical polar coordinates, and then the limits of integration are quite simple, but the integrand gets complicated, unless [itex] n = 1 [/itex], in which case the integral is quite easy to do.

I then thought that, since the only case where this looks simple enough to do directly is [itex] n = 1 [/itex], I could try to make a conjecture as to what the value of the integral is for general n and then try to prove it by induction. The problem with that, though, is that I don't see how to go from the case [itex] n = k + 1 [/itex] to the case [itex] n = k [/itex].

Any ideas?