Further calculus

Reduction formulae, arc length and surface area

A reduction formula links $I_n=\int f_n$ to $I_{n-1}$ or $I_{n-2}$ (usually by parts), so you climb from an easy $I_0$ to any $I_n$. The length of a curve is $\int\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$ and the surface it sweeps out is $2\pi\int y\,ds$.