Line Integral: Step-by-Step Guide
A line integral integrates a function or vector field along a curve C. The scalar form integrates f ds (against arc length); the vector form integrates F dot dr (work done by F along C). For a conservative field F = grad f, the line integral depends only on the endpoints.
Math Racer skill guide · Calculus 3 · College
How to solve it, step by step
- Parametrize the curve. Write r(t) = (x(t), y(t), z(t)) for t in [a, b]. Compute r'(t) and |r'(t)|.\[\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle,\; t\in[a,b]\]
- Set up the integral. For scalar line integral: integral of f ds = integral from a to b of f(r(t)) |r'(t)| dt. For vector line integral: integral of F dot dr = integral from a to b of F(r(t)) dot r'(t) dt.\[\int_C \mathbf{F}\cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}'(t)\,dt\]
- Substitute the parametrization. Replace x, y, z in F (or f) with x(t), y(t), z(t).\[\mathbf{F}(\mathbf{r}(t)) = \langle P(\mathbf{r}(t)), Q(\mathbf{r}(t)), R(\mathbf{r}(t)) \rangle\]
- Compute the dot product or product. Multiply out the integrand into a single function of t.\[\mathbf{F}\cdot\mathbf{r}'(t) = P x'(t) + Q y'(t) + R z'(t)\]
- Evaluate the resulting single-variable integral. Apply standard integration techniques. If F is conservative with potential f, you can shortcut: integral of F dot dr = f(end) - f(start).\[\int_C \nabla f\cdot d\mathbf{r} = f(B) - f(A)\]
Worked example
Evaluate the line integral of F = (y, x) along the curve C parametrized by r(t) = (t, t^2) from t = 0 to t = 1.
Step 1: Parametrize. r(t) = (t, t^2), so r'(t) = (1, 2t).
Step 2: Substitute into F. F(r(t)) = (y, x) = (t^2, t).
Step 3: Compute the dot product F dot r'(t) = t^2 * 1 + t * 2t = t^2 + 2t^2 = 3t^2.
Step 4: Set up integral from 0 to 1 of 3t^2 dt.
Step 5: Evaluate: [t^3] from 0 to 1 = 1.
Verification: F = (y, x) is conservative with potential f(x,y) = xy (since f_x = y, f_y = x). The path goes from (0, 0) to (1, 1), so integral of F dot dr = f(1, 1) - f(0, 0) = 1 - 0 = 1. Matches.
Tips and common mistakes
- Always check whether F is conservative: if curl F = 0 on a simply connected domain, find a potential f and use the Fundamental Theorem of Line Integrals.
- For closed curves and conservative F, the line integral is 0.
- Use Green's theorem for closed curves in the plane: integral of P dx + Q dy = double integral of (Q_x - P_y) dA.
- For scalar line integrals with respect to arc length, you must include |r'(t)|.
- For vector line integrals, do not include |r'(t)| - just dot with r'(t).
- Reversing the direction of C negates a vector line integral but does not change a scalar one.
- Choose the simplest parametrization: line segments r(t) = (1-t)A + tB; circles use cos and sin.
- Check units: a work integral of force along a path has units of force times distance.