I'm trying to find the volume growth of a wedge cut from a sphere in $\mathbb{R}^3$.
In particular the constraints are: $$-log(z+1)^2 \leq x \leq log(z+1)^2,$$ $$-z \leq y \leq z,$$ $$x^2+y^2+z^2\leq r^2.$$
I'm expecting the volume to be of order $\Omega(r^2)$ but I'm struggling with the computation.
The problem where $x$ is bounded by a general smooth, positive, increasing function of $z$ is also of interest. ie $$-f(z) \leq x \leq f(z),$$ $$-z \leq y \leq z,$$ $$x^2+y^2+z^2\leq r^2.$$
I wonder what's an useful technique to deal with these computations?