line integral of a scalar function

A curve-analogue of a definite integral, $\int_C f\,ds$ , that weights a curve by a scalar function and adds up, using arc length $ds$ .

Given $C$ parametrized by $\mathbf{r}(t) = (x(t), y(t))$ for $t \in [a,b]$ , the line integral of a scalar function $f$ over $C$ is $\int_C f\,ds = \int_a^b f(x(t),y(t))\,\sqrt{x'(t)^2+y'(t)^2}\;dt$ , using $ds=|\mathbf{r}'(t)|\,dt$ . Because $ds$ is always positive regardless of which direction $t$ increases, this integral is independent of both the parametrization and the direction of travel — reversing $C$ 's orientation leaves $\int_C f\,ds$ unchanged.

Examples

If $f$ is a wire''s density, $\int_C f\,ds$ is the wire''s total mass.
For the quarter circle $x=\cos t,\ y=\sin t$ , $0\le t\le\pi/2$ , $\int_C (x+y)\,ds = 2$ .