Confusion about average KE in free electron model via Density of States (DOS)

Assume a low temperature regime in which levels up to the Fermi Level, $E_F$, are populated.

I have evaluated the density of states in energy space as $$D(E)=frac{L^3}{pi^2hbar^3}(2m_e^2E)^{1/2},$$

and the Fermi Energy as $$E_F=frac{(3pi^2n_e)^{2/3}}{2m_e}hbar^2,$$

where symbols have usual meanings and $n_e=frac{N}{L^3}$. So far, so good. Now, I wish to evaluate the average kinetic energy of the electrons. Intuitively, I would integrate $$int_0^{E_F}Etimes D(E) dE$$ which would give me the expectation value of kinetic energy $equiv left< E_Kright>$ – or so I thought. Apparently, I need to divide by the number of molecules $N$ once more, such that $$left< E_Kright> = frac{int_0^{E_F}Etimes D(E) dE}{int_0^{E_F} D(E) dE}.$$
I can understand that to find the average energy of an electron, we need to divide through by $N$. But then physically, what is the integral in the numerator above? As far as I remember, for other distribution functions, the expectation value or average of a quantity was only this integral in the numerator. Have I misunderstood something basic? Would be grateful if someone could quickly and logically clear this up for me.