There's this one tantalizing line from the Wikipedia article:
In mathematics, a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points
Can someone expand on that?
Otherwise moments in math don't seem to arise from anything related to the figures of which you find the moments of, but rather are just "nice things to know" about a figure.
E.g. the first moment:
$$ \iint_R y \rho(x,y) dA $$
(That's supposed to be over a region R)
Gets you the first moment with respect to $y$. Which has no meaning in and of itself, but happens to be:
- torque (what it takes to rotate R pushing from the y side)
- the mean (where $\rho$ is a probability function over R)
The second moment gets you the variance (Statistics) and the Moment of Inertia (Physics), etc.
So it looks like this form just "shows up" in various applied disciplines and in math the general form of finding the moments of some figure (a "lamina" as my book calls it) is just:
$$ \iint_R y^n \rho(x,y) dA $$
$$ n = 1, 2, ... $$