Building on the nice answer of Guillaume: The integral

$$ \int_{[-1,1]^n} \prod\_{i < j} |x_i^2 - x_j^2 | dx_1\dots dx_n $$

has the closed-form evaluation

$$ 4^n \prod_{k \leq n} \binom{2k}{k}^{-1}.$$

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:

$$ n! \prod_{k\leq n} \frac{ \pi^k }{ ((k/2)! \binom{2k}{k})} .$$

In particular, we have:

 - 2/3 &pi;<sup>2</sup> for n=2
 - 8/45 &pi;<sup>4</sup> for n=3
 - 4/1575 &pi;<sup>8</sup> for n=4


  [1]: http://en.wikipedia.org/wiki/Selberg_integral