What is the density of states in k-space?

[RadonX]

Homework Statement

I'm having a dumb moment in proving why the following is true:

D(k)d³k=\frac{V}{4\pi^{3}}4\pi k^{2}dk

Homework Equations

NA

The Attempt at a Solution

I realize that the second part 4\pi k^{2}dk is an integration element d³k. But the density of states I can't figure out.
To me it seems that density should be something ON volume. But here it's the other way around.

I'm sure I've done this before and it was fine. Just having a dumb moment that's wasting quite a bit of my time.

 

[RadonX]

I THINK I have it.
For each axis we have a state of a different k value at every;
\frac{2 \pi}{a}n
So rearranging for n and summing up (integrating) over each axis we end up with;
(\frac{a}{2 \pi})^3 \int d^{3} k
The factor of 2 difference between this answer and the one quoted in my notes is due to spin degeneracy.

Does all that sound right?