I was reading about the Kinetic Theory of gases (in Resnick and Halliday) in which I came across The Maxwell-Boltzmann Probability Distribution function for velocities of gases which is defined as:
$$ P(v) = 4\pi (\frac{M}{2\pi RT})^{3/2}v^{2}\exp(-Mv^{2}/2RT)$$
It is said that the probability of a the velocity of a gas lying between the speeds $v_1, v_2$ is given by the improper integral:
$$ \int_{v_1}^{v_2} P(v) dv $$
It is also mentioned that since all the speeds of the gases lie between $0$ and $\infty$, the integral: $$ \int_0^{\infty} P(v) dv = 1$$
I am kinda able to understand the intuition behind these statements but I would like to know why is it that integral from $v_1$ to $v_2$ of the product of $P(v)$ and $dv$ gives us the probability of finding a gas molecule having a velocity between $v_1$ and $v_2$?
Also, it is given that for calculating the average velocity ($v_{avg}$) , it is computed by the integral : $$ \int_0^{\infty} v \ \ P(v) dv $$
Can someone please shed some light on why we are multiplying $P(v)dv$ with $v$ ? Also since we are calculating average velocity, how come we aren't dividing by the number of gas molecules in the container?
I am completely new to how probability distribution curves work, so feel free to assume that I have no prior knowledge on this topic and a detailed, intuitive explanation would be preferred :)