Q.For a given gas molecule at a fixed temperature, the area under the Maxwell-Boltzmann distribution curve is equal to

Why a distribution function is needed at all. Even when a gas sample's macroscopic temperature and pressure are perfectly steady, individual molecules are still constantly colliding and exchanging speeds with each other, so no single molecule keeps a fixed speed for long. What does stay fixed, once the gas reaches thermal equilibrium, is the number of molecules whose speed happens to fall within any chosen range -- this steady population count, as a function of speed, is exactly what the Maxwell-Boltzmann distribution function describes.

The formula. The number of molecules $dN_v$ with speed between $v$ and $v+dv$, out of a total sample of $N$ molecules, is

Graphed against $v$, this curve starts at zero, rises roughly like $v^2$ at low speeds, reaches a single peak exactly at $v_{mp}$, and then falls off sharply (exponentially) at high speeds because of the $e^{-mv^2/2kT}$ factor. The area under a thin strip of width $dv$ at any speed $v$ gives the number of molecules in that narrow speed range, and the total area under the whole curve always equals the fixed total molecule count $N$. …