Line integrals of scalar functions
Problem 12.324 · medium
Evaluate \( \displaystyle \int_C x + y \, ds \), where \( \displaystyle C \) is one turn of the helix r(t) = ⟨cos t, sin t, t⟩.
- Parametrize: r(t) = <cos(t), sin(t), t>, 0 ≤ t ≤ 2*pi; ds = ‖r′(t)‖ dt.Reviewed
- \[ \sqrt{\left(\frac{d}{d t} t\right)^{2} + \left(\frac{d}{d t} \sin{\left(t \right)}\right)^{2} + \left(\frac{d}{d t} \cos{\left(t \right)}\right)^{2}} = \sqrt{2} \]‖r′(t)‖.✓ Proved
- \[ \int\limits_{0}^{2 \pi} \sqrt{2} \left. x + y \right|_{\substack{ x=\cos{\left(t \right)}\\ y=\sin{\left(t \right)}\\ z=t }}\, dt = 0 \]∫ f(r(t)) ‖r′(t)‖ dt.✓ Proved
Answer \( 0 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | f at each segment's midpoint times its length, over a 20,000-segment polyline |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly parametrizes the helix for one turn, computes the arc length element, and evaluates the integral. The result is correct because the integral of cos(t) + sin(t) over [0, 2pi] is zero.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly parametrizes the helix for one turn, computes the arc length element, and evaluates the integral. The result is correct because the integral of cos(t) + sin(t) over [0, 2pi] is zero.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly parametrizes the helix for one turn, computes the magnitude of the derivative vector, and evaluates the resulting integral. The logic is sound and the result is correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly parametrizes the helix for one turn, calculates the arc length element ds, and evaluates the resulting definite integral, which is indeed 0.gpt-oss:20b: pass 2026-10-14
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/line_integral_scalar, checked 2026-10-17 with SymPy 1.14.0.