How is trigonometric substitution done with a triple integral? For instance,
$ 8 \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} (1) dz dy dx $
Here the limits have been chosen to slice an 8th of a sphere through the origin of radius r, and to multiply this volume by 8. Without converting coordinates, how might a trig substitution be done to solve this?