How is it exactly i convert between a k-space sum an integral? Assume that we have some macroscopic solid. Periodic boundary conditions leads to kx,ky,kz = 2π/L, so each k-space state fills a volume (2π/L)^{3} or has a density of V/(2π)^{3}. To then count for instance the number of state with wavevector k<k0, what do you then do? Intuitively I would multiply the volume of a cube of radius k0, but how does this translate into an integral exactly?
You can also write the sum as an integral over a sum of delta functions. For slowly varying test functions, the delta functions may then be replaced by their density ##V/(2\pi)^3##.