What is contour?

A path is a piecewise $\mathcal C^1$ map $\gamma \colon \intcc ab \to \C$; it is closed if $\gamma(a) = \gamma(b)$. For continuous $f$ on the image of $\gamma$:

(the ML inequality; length $= \int_a^b\abs{\gamma'}$). The integral is invariant under increasing $\mathcal C^1$ reparametrization and changes sign under orientation reversal.