I am interested in computing the volume of the following region, as a function of $X$:
$$\displaystyle \{(x,y,z) \in \mathbb{R}^3 : |x|, |y|, |z^2 - 4xy| \geq 1, |xy(z^2 - 4xy)^2| \leq X\}.$$
This region is bounded and closed, hence compact, so it must have finite volume. Does anyone know how to do the calculation?
The region is symmetric with respect to the joint symmetry operation $x \to -x$ and $y \to -y$, and $z$ appears only as a square, so there are only two octants you need integrate, e.g., $x>0, y>0, z>0$ and $x>0, y<0, z<0$.
Incidentally, here is the region for $X = 4000$:
It is an ugly integral, but simple to perform numerically for fixed $X$ (and which gives for $X = 4000$ the value $2813$).
I'd be interested in how this problem arose.