Line integrals
Objective. Evaluate scalar and vector line integrals.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Line integrals extend integration to curves, computing ∫_C f ds for scalar fields or ∫_C F·dr for vector fields along a parametrised path r(t), a ≤ t ≤ b. For a scalar field, ∫_C f ds = ∫ₐᵇ f(r(t)) |r'(t)| dt, while for a vector field, ∫_C F·dr = ∫ₐᵇ F(r(t))·r'(t) dt. These compute quantities like work done by a force along a path. For F = (y, −x) along the straight path r(t) = (t, t), 0 ≤ t ≤ 1, F(r(t))·r'(t) = (t)(1) + (−t)(1) = 0, so the line integral is zero despite F not being conservative.
Worked example
Evaluate the line integral of F = (x, y) along the path r(t) = (cos t, sin t) for 0 ≤ t ≤ π/2.
- 1Compute r'(t) = (−sin t, cos t).
- 2Compute F(r(t))·r'(t) = (cos t)(−sin t) + (sin t)(cos t) = 0.
- 3Integrate the zero integrand over [0, π/2]: ∫₀^{π/2} 0 dt = 0.
- 4So the work done by F along this quarter-circle path is zero.
Answer: 0
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Start practisingCommon mistake. Students forget to use the speed |r'(t)| rather than r'(t) itself when integrating scalar fields over a curve, confusing arc-length integrals with vector-field work integrals.
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