Line Integrals of Functions

Main idea

A line integral $\int_C f\,ds$ adds up a quantity along a curve, piece by piece of length. Think of a bent wire whose density changes along it: its mass is the density times each tiny length, all added up.

\int_C f\,{\htmlData{sym=0}{ds}} = \int_a^b f\big(\vec r(t)\big)\,\|\vec r\,'(t)\|\,dt

What the symbols mean

$ds$
Arc length element
‖r′(t)‖ dt: a tiny piece of the curve's length.

Explanation

A line integral of a function is simply a way to add up something that lives along a curve instead of along the $x$ -axis: the mass of a bent wire, the charge on a curved cable, the total heat lost along a pipeline. In $\int_C f\,ds$ , $C$ is the curve, $ds$ is a tiny piece of its length and $f$ is how much stuff there is per unit length at that spot. Multiply and add. The formula with $\|\vec r\,'(t)\|$ in it looks heavier, but it's only telling you how to measure those tiny lengths.

Take a look at an ordinary integral $\int_a^b f(x)\,dx$ . You're walking along the $x$ -axis, and $dx$ is the length of each step. A line integral is the same walk along a curve, so the steps are pieces of arc length $ds$ instead.

Where the speed comes from. You describe the curve with a parametrization $\vec r(t)$ , and as $t$ moves a little, by $dt$ , the point moves a distance of about $\|\vec r\,'(t)\|\,dt$ : speed times time. That's the same $ds$ you used for arc length in Calculus 2, $\sqrt{x'(t)^2 + y'(t)^2}\,dt$ . So:

Recipe: parametrize the curve, substitute $x(t), y(t)$ into $f$ , multiply by the speed $\|\vec r\,'(t)\|$ , and integrate over the parameter interval.

With $f = 1$ you get the curve's length. With $f$ a density, you get the mass of a wire. Divided by the length, it's the average value of $f$ along the curve.

The answer doesn't depend on which direction you traverse $C$ , or on which parametrization you choose. It makes sense: a wire has the same mass whichever end you start weighing from. If you trace the curve twice as fast, $dt$ covers twice the length, and the speed factor exactly makes up for it.

A common mistake. It's tempting to substitute into $f$ and integrate $dt$ , forgetting the speed. That only works when the speed is $1$ . On the circle $\langle 4\cos t, 4\sin t\rangle$ the speed is $4$ , and leaving it out makes the answer four times too small.

See it

A curtain hung along the quarter circle C, as tall as f(x, y) = 1 − xy/2 at each point. Each sliver is f high and ds wide, so the curtain's area is ∫ f ds. Notice the curtain is tallest in the middle of the arc, height 2, and only 1 at the ends, so the area grows fastest mid-sweep, finishing at π + 2 ≈ 5.14. Drag to turn it.

Worked examples

Example 1

Evaluate $\int_C xy^4\,ds$ , where $C$ is the right half of $x^2 + y^2 = 16$ .

1
Parametrize the right half as $\vec r(t) = \langle 4\cos t, 4\sin t\rangle$ , $-\frac\pi2 \le t \le \frac\pi2$ . Its speed is $\|\langle -4\sin t, 4\cos t\rangle\| = 4$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find $\int_C 1\,ds$ along the segment from $(0, 0)$ to $(3, 4)$ .

Problem 2+5 XP

Evaluate $\int_C(x + y)\,ds$ along the segment from $(0, 0)$ to $(1, 1)$ .

Problem 3+5 XP

A wire along the unit circle has constant density 2. Find its mass.

Problem 4+5 XP

Evaluate $\int_C x\,ds$ along $y = x^2$ from $(0, 0)$ to $(1, 1)$ .

Problem 5+5 XP

Evaluate the example's integral, $\int_C xy^4\,ds$ on the right half of $x^2 + y^2 = 16$ .

Problem 6+5 XP

A 5 m steel cable runs straight from $(0, 0)$ to $(3, 4)$ (in meters). Its density rises steadily from 2 kg/m at the first end to 4 kg/m at the other. Find its mass, in kilograms.

Problem 7+5 XP

A thin wire bent into the upper half of the unit circle has constant density. Find the $y$ -coordinate of its center of mass.

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A spring has the shape of the helix $\vec r(t) = \langle\cos t, \sin t, t\rangle$ , $0 \le t \le 2\pi$ , and its density is $\rho = z$ , so it gets heavier toward the top. Find its mass and the $z$ -coordinate of its center of mass. Enter both.

Where it's used

The mass and center of mass of cables, the total charge on a curved wire, and the average temperature along a pipeline are line integrals of functions.

Vector Fields Line Integrals of Vector Fields