<s>[INST] Any solution for $\iiint\frac{x^2+2y^2}{x^2+4y^2+z^2}\,dv$ I tried to solve this triple integral but couldn't integrate the result.
$$\iiint\frac{x^2+2y^2}{x^2+4y^2+z^2}\,dv$$ and the surface to integrate in is $$x^2+y^2+z^2\le1$$
Is there any way to transform the integral into polar coordinates?
 [/INST] Trivial by symmetry: the unit ball $B\subset\mathbb{R}^3$ is invariant under the transformation $(x,y,z)\mapsto(z,y,x)$, hence:
$$\iiint_B \frac{x^2+2y^2}{x^2+4y^2+z^2}\,d\mu = \frac{1}{2}\left(\iiint_B \frac{x^2+2y^2}{x^2+4y^2+z^2}\,d\mu+\iiint_B \frac{z^2+2y^2}{x^2+4y^2+z^2}\,d\mu\right) = \frac{1}{2}\mu(B) = \color{red}{\frac{2\pi}{3}}.$$
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