Does this have something to do with definite integration?
Yes, The Maxwell Boltmann distribution (for classical particles) or more generally the Boltzmann distribution denotes the socalled "density of states (DOS)". This quantity expresses how close consecutive electronical energylevels are spearated from each other. High DOS means many energylevels in an energy-interval [E, E+dE]. If you integrate over this interval (this is a definite integration indeed) you get the total number of particles (ie electrons) in that specific energy interval.
The DOS and PDOS (partial DOS) are very important quantities for studying the influence of defects (like missing atoms in crystals) onto the electrostatics of many particle bodies like crystals.
Yes, it does. Whoever you are quoting there, however, only had the 1-dimensional distribution in mind. In fact, the Maxwell-Boltzmann distribution applies also to a collection of particles with velocities in three dimensions. The integrations are over all velocities so that if f(v) is the distribution function then
[itex]\int f(\vec v) dv_x dv_y dv_z = n[/itex]
is the number density of particles in real space. Strictly speaking, it is not an area. Generally, [itex]f = f(\vec x, \vec v, t)[/itex] allowing for both temporal and spatial variation of the distribution function.
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