Line Integrals

Line integrals generalize ordinary integrals to integration along curves, computing quantities such as work done by a force field along a path or the mass of a wire with varying density.

Scalar Line Integrals

Definition

Let $C$ be a smooth curve parametrized by $\mathbf{r}(t) = (x(t), y(t), z(t))$ for $a \leq t \leq b$ , and let $f$ be a continuous scalar function on $C$ . The scalar line integral of $f$ along $C$ is $\int_C f\,ds = \int_a^b f(\mathbf{r}(t)) \|\mathbf{r}'(t)\|\,dt$ where $ds = \|\mathbf{r}'(t)\|\,dt$ is the arc length element. This integral is independent of the parametrization (and its orientation).

Vector Line Integrals

Definition

Let $\mathbf{F}$ be a continuous vector field and $C$ an oriented smooth curve parametrized by $\mathbf{r}(t)$ , $a \leq t \leq b$ . The vector line integral (or work integral) of $\mathbf{F}$ along $C$ is $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt = \int_C P\,dx + Q\,dy + R\,dz$ This measures the work done by $\mathbf{F}$ on a particle moving along $C$ . Reversing orientation changes the sign: $\int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_C \mathbf{F} \cdot d\mathbf{r}$ .

Examples

ExampleWork along a helix

The helix $\mathbf{r}(t) = (\cos t, \sin t, t)$ for $0 \leq t \leq 2\pi$ under the force $\mathbf{F} = (y, -x, 1)$ : $\mathbf{r}'(t) = (-\sin t, \cos t, 1)$ $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} (\sin t)(-\sin t) + (-\cos t)(\cos t) + 1\,dt = \int_0^{2\pi} (-1 + 1)\,dt = 0$

Wait: $(\sin t)(-\sin t) + (-\cos t)(\cos t) + 1 = -\sin^2 t - \cos^2 t + 1 = 0$ . So $\int_C \mathbf{F} \cdot d\mathbf{r} = 0$ .

ExampleCirculation around a circle

For $\mathbf{F} = (-y, x)$ and $C$ the unit circle $\mathbf{r}(t) = (\cos t, \sin t)$ , $0 \leq t \leq 2\pi$ : $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} [(-\sin t)(-\sin t) + (\cos t)(\cos t)]\,dt = \int_0^{2\pi} 1\,dt = 2\pi$

RemarkPhysical interpretation

The scalar line integral $\int_C f\,ds$ computes the total mass of a wire with linear density $f$ , or the total value of $f$ accumulated along $C$ . The vector line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ computes the work done by the force $\mathbf{F}$ (the component of $\mathbf{F}$ tangent to $C$ , integrated over arc length).