I came across this article which had this photo:
Ok, here is the quiz:
What are these formulas?
Derive one of these formulas
Derive two of these formulas (hint: one way to derive one of these involves a change of variables and polar coordinates.
Assuming you have a piecewise continuous function (ok, make it piecewise smooth if you wish) and is periodic over derive the middle formula.
You win: all of the money I made writing this note. 😉