Line Integrals: Work, Circulation and Flux Explained

Pushing a lawnmower around a curved garden path, you do work — but the force and the direction of motion both keep changing along the way. A single number like “force times distance” cannot capture that. You need to add up the tiny contributions of force along the direction of motion at every point of the path, and that is precisely what a line integral does.

There are two main kinds. The line integral of a scalar function, ∫C f ds, adds up values of f weighted by arc length — used for the mass of a wire with variable density. The line integral of a vector field, ∫C F·dr, adds up the component of F along the path’s direction — this is the work done by a force field F along the curve C. Both are computed by parametrizing the curve as r(t) and integrating from the start to the end of the parameter.

Key Concepts

• Parametrization: r(t) = (x(t), y(t)), a ≤ t ≤ b, traces the curve C; dr = r′(t) dt.
• Scalar line integral: ∫C f ds = ∫ab f(r(t)) |r′(t)| dt.
• Vector line integral (work): ∫C F·dr = ∫ab F(r(t))·r′(t) dt.
• Orientation matters for F·dr: reversing the curve flips the sign; for f ds it does not.
• If F = ∇f (conservative), then ∫C F·dr = f(end) − f(start) — path independent (Fundamental Theorem for Line Integrals).

Formulas

• ∫C F·dr = ∫ab (P dx + Q dy) = ∫ab [P(x(t),y(t)) x′(t) + Q(x(t),y(t)) y′(t)] dt
• Work = ∫C F·dr.
• Circulation: ∮C F·dr around a closed loop.

Worked Example

Find the work done by F(x, y) = (y, x) on a particle moving along r(t) = (t, t²) from t = 0 to t = 1.

1. Field along the curve. F(r(t)) = (t², t).
2. Tangent vector. r′(t) = (1, 2t), so dr = (1, 2t) dt.
3. Dot product. F(r(t))·r′(t) = t² · 1 + t · 2t = t² + 2t² = 3t².
4. Integrate. ∫01 3t² dt = [t³]01 = 1 − 0 = 1.
5. Answer: 1 unit of work. Cross-check: F = ∇(xy), so the field is conservative and work = xy at (1,1) minus xy at (0,0) = 1 − 0 = 1. It agrees.

Common Misconceptions

• “∫C F·dr is the same as ∫C f ds.” The vector version uses the dot product with the direction of travel; the scalar version uses arc length and ignores direction.
• “The parametrization must use arc length.” Any smooth parametrization works — the |r′(t)| or r′(t) factors automatically correct for speed.
• “Work is always positive.” If the field opposes the motion, the dot product — and the work — is negative.

Real-World Applications

• Physics: work done by gravity, electric, or magnetic forces along a path.
• Aerodynamics: circulation of fluid around an airfoil, linked to lift.
• Mass of a curved wire: ∫C (density) ds.
• Electromagnetism: voltage around a loop; magnetic circulation (Ampère’s law).

A line integral accumulates a quantity along a curve instead of along an axis: parametrize the path, substitute into the integrand, and integrate the parameter from start to finish. Use f ds when direction does not matter (mass of a wire) and F·dr when it does (work) — and watch the orientation, because walking the path backwards flips the sign of work.