What does the area under Fermi Distribution curve represent? I am struggling to understand what does the area under the fermi distribution curve represent when plotted like this:
I know that area under the Maxwell Distribution curve when plotted for speeds of molecule represent total number of molecules and it's supposed to stay constant when temperature is increased.
I understand it as Y-axis represent number of molecules per unit speed and X-axis represent speed, so naturally the area under curve would give total number of molecules. So my question is: Does the area under Fermi curve represent the total number of electronic states? And will it remain constant upon increasing temperature?
 A: The Fermi distribution represents the probability of a state at the given energy to be occupied.
For example, at $T=0\,\mbox{K}$, all the states with energy lower than the Fermi energy $E_F$, i.e., the chemical potential at zero temperature $E_f = \mu(T=0)$, are occupied (indeed from the plot we can see that the probability ob being occupied is $1$).
In my opinion, rather than looking for a physical interpretation for the integral of the Fermi-Dirac distribution, it is better to see how we compute observables from it.
In order to do that, you have to introduce another quantity: the density of states $g(E)$, which represents the numeric density of allowed states at a given energy in a unit volume of your sample (usually in solid-state physics all quantities are normalized for the volume). More precisely
\begin{equation}
g(E) = \frac{1}{V} \frac{\mathrm{d}N_{states}}{\mathrm{d}E}
\end{equation}
The density of states is a function of the energy that depends on the system you are considering.
For instance, if you want to compute the internal energy $u$ per unit volume of your system, you have to compute first the states actually occupied. You can achieve this by multiplying the density of states $g(E)$ for the probability of being occupied, that is, the Fermi-Dirac distribution $f(E)$. Once you have found the density of states occupied by a particle, you sum all the energies of such states. Practically you do this with an integral over all possible energies. In a formula:
\begin{equation}
u=\frac{U}{V} = \int \epsilon\, \underbrace{f(\epsilon)\, g(\epsilon)}_{\mbox{DOS occupied}}\,\mathrm{d}\epsilon
\end{equation}
In a simpler case, if you want to compute the total number of particles per unit volume of your system, you have to integrate (=sum) the number of occupied states over all possible energies.
\begin{equation}
n = \frac{N}{V} = \int f(\epsilon)\, g(\epsilon)\,\mathrm{d}\epsilon
\end{equation}
A: Fermi Dirac function is a distribution function or weighing function (just like you would use mass as weight factors for coordinates when you calculate Centre of Mass; or use a weight factor of 1 when calculating the coordinates of centroid of a geometry). It is the weight factor that you expect when you have fermions, more precisely in fermion statistics. Mathematically this comes under a class of functions called Probability density function.
To understand what integral of (or equivalently limit of sums or vaguely sum of) this means, we start with asking what Fermi Distribution is.
Here is a very simple derivation which should explain what it means. Stating it explicitly, Fermi distribution function gives the average value for an occupancy number(i.e. occupancy averaged over all states) where the total occupancy number is normalized to one (which is part of definition of Probability density function, for our case its sufficient to say that this normalization is just a convention to avoid any factors that might lurk around otherwise).
Hence integral (or sum if you are following the derivation of Fermi function using Canonical Ensemble) of Fermi function simply represents the sum of average  occupancy number over states which is essentially the total occupancy number, which by convention we conveniently normalized it to 1, essentially satisfying the conditions for a Probability density function.
$$ \int f(\epsilon)\, \mathrm{d}\epsilon=1$$
To summarise,
Fermi Distribution is just a weight function. It is important because this is what differentiate the various statistics (fermi, Bose or Maxwell's), in other words, this particular distribution captures the inherent property of the system (namely its composition) and does only that. You have to multiply it with Density of States (DOS) to make a better physical sense, as @Davide stated.
