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Now I know that the general recipe for this is:

[tex] n = \int{N(E).dE} = \int{D.dE} = DE [/tex]

So that the energy is [tex] E = \frac {n}{D} [/tex] and this is the right answer for the Fermi energy. I'm just not sure why I know that this energy is the Fermi energy. I know it has something to do with the fact that at T = 0 the max energy is the fermi energy, but I'm not sure how to link that to this problem?

Also does anyone know how I get from the above result to the total energy of the Fermi gas, and how this stuff relates to thermal velocity in metals?