= Solution
In two dimensions, each allowed wavevector occupies area $(2\pi)^2/A$ in $k$-space. Including the two spin states, the number of states in an annulus $(k,k+dk)$ is
$$
d\mathcal N
=2\frac{A}{(2\pi)^2}2\pi k\,dk.
$$
Since
$$
E=\frac{\hbar^2k^2}{2m},
\qquad
k\,dk=\frac{m}{\hbar^2}\,dE,
$$
the <two-dimensional free-electron density of states> is
$$
\boxed{g(E)=\frac{d\mathcal N}{dE}=\frac{Am}{\pi\hbar^2}}
$$
for $E\geq0$. The <Fermi-Dirac distribution> is
$$
\boxed{f(E)=\frac1{e^{\beta(E-\mu)}+1}},
\qquad \beta=\frac1{k_BT}.
$$
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