I have the integral of a function $f(x,y,z)$ defined over the region $a<\sqrt{x^2-y^2-z^2}<b$; that is, $$I\equiv \int_{a<\sqrt{x^2-y^2-z^2}<b}dxdydzf(x,y,z).$$ I realized that defining the variables $$x=\rho \cosh \chi,\\ y =\rho\sinh \chi \cos \varphi,\\ z = \rho\sinh \chi \sin \varphi,$$ implies $a<\rho<b$, so $$I=\int_a^b d\rho \int_{?_1}^{?_2} d\chi \int_0^{2\pi}d\varphi \rho^2\sinh \chi f(\rho \cosh \chi,\rho\sinh \chi \cos \varphi,\rho\sinh \chi \sin \varphi)$$ The factor $\rho^2 \sinh \chi$ corresponds to the Jacobian determinant.
How can I obtain the limits of integration for $\chi$? How is this done in general?