Line integrals of scalar functions
Problem 12.416 · medium
Evaluate \( \displaystyle \int_C x^{2} + y^{2} + z \, ds \), where \( \displaystyle C \) is the quarter circle of radius 3 in the xy-plane, from (3, 0, 0) to (0, 3, 0).
- Parametrize: r(t) = <3*cos(t), 3*sin(t), 0>, 0 ≤ t ≤ pi/2; ds = ‖r′(t)‖ dt.Reviewed
- \[ \sqrt{\left(\frac{d}{d t} 0\right)^{2} + \left(\frac{d}{d t} 3 \sin{\left(t \right)}\right)^{2} + \left(\frac{d}{d t} 3 \cos{\left(t \right)}\right)^{2}} = 3 \]‖r′(t)‖.✓ Proved
- \[ \int\limits_{0}^{\frac{\pi}{2}} \left. x^{2} + y^{2} + z \right|_{\substack{ x=3 \cos{\left(t \right)}\\ y=3 \sin{\left(t \right)}\\ z=0 }} \cdot 3\, dt = \frac{27 \pi}{2} \]∫ f(r(t)) ‖r′(t)‖ dt.✓ Proved
Answer \( \frac{27 \pi}{2} \)
✓ 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 parametrization correctly models the quarter circle in the xy-plane with the correct orientation and radius. The calculation of the arc length element and the subsequent integration are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The parametrization correctly models the quarter circle in the xy-plane with the correct orientation and radius. The calculation of the arc length element and the subsequent integration are correct.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The parametrization correctly models the quarter circle in the xy-plane with the correct orientation and bounds. The calculation of ds and the subsequent integration are correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The parametrization correctly models the quarter circle in the xy-plane with the correct orientation and radius. The calculation of the arc length element and the resulting integral are correct.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.