Line Integrals

_C f ds (arc-length) and _C F · d r (work) — parameterize, dot, integrate.

The idea

A line integral adds up a function along a curve $C$ that bends through the plane or through space, as the ordinary integral $\int_a^b f(x)\,dx$ adds one up along a straight interval.

There are two versions, and they differ only in what each small piece of the curve is weighted by. Chop $C$ into short arcs. Weight each arc by its own length and by the value of a scalar $f$ there, add, and take the limit; the result is the scalar line integral $\int_C f\, ds.$ If $f$ is the density of a bent wire, this integral is the wire's mass. Weight the arcs instead by their displacement — a small vector — dotted against a vector field $\mathbf{F}$, and the limit is $\int_C \mathbf{F} \cdot d\mathbf{r}.$ The dot product keeps only the part of $\mathbf{F}$ that points along the direction of travel, so if $\mathbf{F}$ is a force, this integral is the work it does on a particle carried along $C$.

To compute either one, describe $C$ by a parameterization $\mathbf{r}(t)$, $t \in [a, b]$. Substituting it turns both line integrals into ordinary single integrals in $t$.

Ways to work on it

• Walkthrough. Both flavors of line integral; compute the work of a vector field around a closed curve.
• Practice. Work of a constant field along a line segment — and why constant fields are conservative.
• Hardest. A scalar line integral over a segment — speed, arc length, and orientation-independence.

Not sure where to start? Take the ten-question placement test.