How Do These Expressions in Statistical Mechanics Equate?

[Appledave]

I was reading the solution to a statistical mechanics problem and this showed up:

http://imageshack.us/photo/my-images/196/grddar.jpg/

S_2N-1 = the area of the 2N-1 dimensional unit sphere.

Could anyone shed some light on how these expressions equal each other, I am quite dumbfounded :(.

 

[xaos]

my guess is that, assuming your domain of integration is a high dimensional 2n-ball and you're integrating over all of R^2n, setting the radius=1 yields the (2n-1)-volume of a fixed boundary, which in (2n-1)+1 spherical coordinates is a S^(2n-1) sphere. then product this with the remaining radial dimension you get back the original ball integral.