Topic 6 of 7

Line and surface integrals

Step 18 · Multivariable calculus topics

Integrate along curves and over surfaces: add up a function along a path, find the work a force field does, and measure the flow through a surface.

Line integrals of functions

To add up f along a curve C, describe the curve as r(t) for a ≤ t ≤ b. Each small piece has length ds = |r′(t)| dt.

\int_C f\,ds = \int_{a}^{b} f(\mathbf{r}(t))\,|\mathbf{r}'(t)|\,dt

Example: A segment

Find the integral of f(x, y) = x + y along the segment from (0, 0) to (3, 4).

r(t) = ⟨3t, 4t⟩ for 0 ≤ t ≤ 1, with |r′| = 5.
$\int_{0}^{1} 7t \cdot 5\,dt = \frac{35}{2}$ .

Answer17.5

Example: Mass of a wire

A wire bent into the quarter circle of radius 2 in the first quadrant has density x. Find its mass.

r(t) = ⟨2 cos t, 2 sin t⟩ for 0 ≤ t ≤ π/2, with |r′| = 2.
$\int_{0}^{\pi/2} 2\cos t \cdot 2\,dt = 4$ .

Answer4

Work and line integrals of vector fields

A vector field F gives an arrow at every point, such as a force or a flow. The work done by F along C adds up the part of F along the path, F · dr.

\int_C \mathbf{F} \cdot d\mathbf{r} = \int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt

The field F = ⟨−y, x⟩ swirls counterclockwise. Around the circle of radius 2 it always pushes along the path, so it does positive work.

Example: A swirling field

Find the work done by F = ⟨−y, x⟩ once around the circle of radius 2, counterclockwise.

r(t) = ⟨2 cos t, 2 sin t⟩, so r′ = ⟨−2 sin t, 2 cos t⟩.
F · r′ = 4 sin²t + 4 cos²t, which is 4.
Integrate 4 from 0 to 2π.

Answer8π

Example: A gradient field

F = ∇φ with φ = x²y. Find the work along any path from (0, 0) to (2, 3).

For a gradient field, the work is φ at the end minus φ at the start.
φ(2, 3) − φ(0, 0) is 12 − 0.

Answer12, whatever the path

Surface integrals and flux

The flux of F through a surface S measures how much flows across it: add F · n dS, where n is the unit normal. Only the part of the flow straight through the surface counts.

\iint_S \mathbf{F} \cdot \mathbf{n}\,dS

Example: A flat square

F = ⟨0, 0, 3⟩ flows through the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 in the plane z = 1, with n = ⟨0, 0, 1⟩. Find the flux.

F · n = 3 everywhere on the square.
The square's area is 4.

Answer12

Example: A sphere

Find the flux of F = ⟨x, y, z⟩ outward through the sphere of radius 2.

On the sphere, n = ⟨x, y, z⟩ ÷ 2, so F · n = (x² + y² + z²) ÷ 2, which is 2.
The sphere's area is 16π.

Answer32π

Try it yourself

Work each one on paper first, then open the answer to check.

Find the length of the segment from (0, 0) to (6, 8) as the line integral of 1.
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10
F = ⟨1, 0⟩. Find the work along the segment from (0, 0) to (5, 0).
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5
F = ⟨1, 0⟩. Find the work along the segment from (0, 0) to (0, 5).
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0: the force is at right angles to the path
F = ∇(xy). Find the work from (1, 1) to (3, 2).
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5
What is the work done by a conservative field around a closed loop?
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0
F = ⟨0, 0, 2⟩ flows through the unit square in the plane z = 0, with n = ⟨0, 0, 1⟩. Find the flux.
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2