I have a question which has been driving me crazy for a while now. I've been googling to find an answer but with no success.
I am aware of the "Mean Value Theorem" for integrals but have seen in some physics books that the mean value is expressed in the form : $F_{Mean}=\frac{\int f(x) dx}{\int dx}$ (not exactly but similarly).
Also, the other day I was discussing "particle size distribution" with my supervisor and he said the average particle size that follows the following distribution $N=C\int_{r_0}^{r_1} r^{-q}$ can be found by: $$<r>=\frac{\int N (r) . r dr}{\int N(r) dr}$$
Could anyone explain where does the notion of "integral division: come from? What is it called?I've been googling different keywords to read on the topic but no success. I can see the similarity between $F_{Mean}=\frac{\int f(x) dx}{\int_a^b dx}$ and $F_{Mean}=\frac{1}{(b-a)}\int f(x) dx$ but in text books I have only seen the latter notion of the MVT. Are those the same thing? How about the equation for the average particle size?
Any help will be very much appreciated!