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.
The condition on z for a point to lie in U is , where The integral for negative z is the same as for positive z, and so
(doing the z integral first).
The variables conveniently separate and give you Check that For the other integral, substitute and get the integral in the form
which you can easily integrate. I get the answer for the measure of U to be (but don't trust my accuracy there). Note that this becomes infinite when k=2. I guess that U must have infinite measure whenever .