How to compute the volume of intersection between two hyperspheres Let's say I have two n-spheres and I've no prior knowledge about the spheres (such as one of the sphere might be inside the other one) and I need to compute the volume of the intersection of the two hyper-spheres. Is there an efficient and generic way to compute the hyper-spherical cap of the intersection of these n-spheres? Note that the hyperspheres are expected to be very high dimensional such as 4096.

It seems to me that the answer written above by Matt is incorrect. Specifically, the expression that is believed to be the exact value of $V_n^{cap}(r,a)$ is in fact only an underestimation which gets farther from the correct value as $n$ is increased. 
Formally the radius of $V_n^{cap}$ for a specific $x$ should be 
$$\sqrt{r^2-(r-x)^2}=\sqrt{2rx-x^2},$$
instead of just $x$.
Thus, the integral for this expression should look as follows
$$V_n^{cap}(r,a)=\int_a^r V_{n-1}\left(\sqrt{2rx-x^2}\right)dx,$$
although I'm not sure how it can integrated.
