The question is pretty much in the title. We were also given the hint that it could be useful to use spherical coordinates when calculating the integral (the actual answer is not required, just its form as a standard integral). The Riemannian metric we were given is: $$G=4R^2\frac{du^2+dv^2+dw^2}{(1+u^2+v^2+w^2)^2},$$ where $(u,v,w)$ are stereographic coordinates.
Any help is much appreciated!
Let me give you a sketch indicating how to proceed.
Your goal is to find the volume of a coordinate patch $(U,\phi)$ in a Riemannian manifold $M$. In those coordinates, write your metric as a $3\times 3$ matrix $g = (g_{ij})$. Recall that in local coordinates, the volume form is given by $$dVol = \sqrt{\det g}dx\wedge dy\wedge dz.$$ Now to find the volume of $U$, integrate the volume form over $U$.
If $M-U$ has measure zero, then what can you say about $Vol(M)$ and $Vol(U)$?