Originally Posted by

**ymar** I have to calculate the Lebesgue measure of

$\displaystyle \left\{(x,y,z)\,:\,x^2+y^2\leq 1,\,(1-x^2-y^2)^{k}z^2\leq(x^2-y^2)^2\right\}$

for every $\displaystyle k\in\mathbb{Z}$

After the substitution I get $\displaystyle \int_U r drd\alpha dz,$ where

$\displaystyle U=\left\{0<r<1,\,\alpha\in(0,2\pi),\,(1-r^2)^kz^2<r^4\cos^{\color{red}2}2\alpha\right\}.$ Note the square there!

I can't calculate it. Everything I try leads to difficult integrals. Could you please give me an idea of what the right way of doing this is?