I have to calculate the Lebesgue measure of

\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

After the substitution I get where

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?

PS: Latex wouldn't work, so I left the code. If any moderator could fix this, I would be very grateful.