Change of Variables & the Jacobian · Superchallenge problem

Find the volume of the solid in the first octant bounded by the six parabolic cylinders $y = x^2$, $y = 2x^2$, $z = y^2$, $z = 2y^2$, $x = z^2$ and $x = 2z^2$; that is, the set of points with $x, y, z > 0$, $x^2 \le y \le 2x^2$, $y^2 \le z \le 2y^2$ and $z^2 \le x \le 2z^2$.