The Band structures page shows the derivation for the amount of states in a filled band, but I do not understand how the integration over the Brillouin zone is done, any help would be much appreciated
Try figuring out what to provide as integration limits for the integrals over k_x, k_y, and k_z. Compare also with the size of the 1D Brillouin zone.
The bounds would be -pi/a to pi/a, but would we then not use spherical coordinates for the integration? and then why not?
The 3D Brilloiun zone is a cube, not a sphere.