Building on the nice answer of Guillaume: The integral
\int<sub>[-1,1]<sup>n</sup></sub> \prod<sub>i<j</sub> |x<sub>i</sub><sup>2</sup> - x<sub>j</sub><sup>2</sup>| dx<sub>1</sub>...dx<sub>n</sub>
has the closed-form evaluation
4<sup>n</sup> / \prod<sub>k≤n</sub> \binom{2k}{k}.
This basically follows from the evaluation of the [Selberg beta integral][1] S<sub>n</sub>(1/2,1,1/2).
Combined with modding out by a typo, we now arrive at the following product formula for the volume of the unit ball of nxn matrices in the matrix norm:
n! \prod<sub>k≤n</sub> π<sup>k</sup> / ((k/2)! \binom{2k}{k}).
In particular, we have:
- 2/3 π<sup>2</sup> for n=2
- 8/45 π<sup>4</sup> for n=3
- 4/1575 π<sup>8</sup> for n=4
[1]: http://en.wikipedia.org/wiki/Selberg_integral