Line Integrals

We are given a smooth space curve $C$ defined by the parametric equations $x=x(t)$, $y=y(t)$, $z = z(t)$, for $a \leq t \leq b$, and a function $f$ which is defined on $C$. Sometimes we work with a vector function $\mathbf{r}$ instead of the parametric equations:

$$\mathbf{r}(t)=\left\langle x(t), y(t), z(t)\right\rangle, \mbox{ for }a \leq t \leq b.$$