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Line Integrals: integrating functions over scalar fields or vector fields

Line integral over a scalar field
\int\limits_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt.
Line integral over a vector field


\int\limits_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt.