An Integral With an Odd Function That Isn't Contrived

Question

When ever I teach calculus, single or multivariable, there is always the point in the text when the author covers odd functions and then gives an example of an integral to evaluates to $0$ because the function inside is odd with respect to one of the variables. While this is neat, it just seems to be time spent on a subject for the sake of one example that isn't all too useful.

My question is: Are there any examples of a situation where the integral over a symmetric region of an odd function show up? I would prefer an example where the function comes about naturally. In other words, not something like "Suppose your distance from the grocery store is given by (odd function). . ."

Answer 1

Symmetry arguments occur all the time, and are incredibly useful. For example, in Fourier series (where calculating the coefficients requires integrating your function times sines or cosines), knowing that your function is even or odd can immediately take care of half the coefficients.

EDIT: Another application: in probability and statistics, for a density that is symmetric, the odd central moments are all $0$.

Answer 2

This sort of symmetry argument shows up in center of mass applications or, by extrapolation, in applications of Green's Theorem or the Divergence Theorem that result in integrals where symmetry is helpful.