The last step is multiplied by two is because the integrand in the first integral is even and the bounds of integration are symmetric about the origin, so the integral can be rewritten as from 0 to k D after scaling by a factor of 2. This is also equivalent to the statement that the volume of a one-dimensional ball is twice its radius. Applying a change a substitution of k = ω ω v s , our bounds are now 0 to ω ω D = k D v s , which gives us our rightmost integral. We continue;