Line integrals of scalar functions
Problem 12.310 · medium
Evaluate \( \displaystyle \int_C x y + z \, ds \), 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
- \[ \int\limits_{0}^{2 \pi} \sqrt{2} \left. x y + z \right|_{\substack{ x=\cos{\left(t \right)}\\ y=\sin{\left(t \right)}\\ z=t }}\, dt = 2 \sqrt{2} \pi^{2} \]∫ f(r(t)) ‖r′(t)‖ dt.✓ Proved
Answer \( 2 \sqrt{2} \pi^{2} \)
✓ 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, computes the arc length element, and evaluates the integral. The bounds 0 to 2*pi correspond to one full turn as requested.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly parametrizes the helix, computes the arc length element, and evaluates the integral. The bounds 0 to 2*pi correspond to one full turn as requested.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, and sets up the line integral with the correct integrand and bounds. The final result is correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly parametrizes the helix, computes the magnitude of the derivative vector, and sets up the scalar line integral with correct bounds and integrand substitution.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.