Line integrals of scalar functions
Problem 12.408 · medium
Evaluate \( \displaystyle \int_C x + y + z \, ds \), where \( \displaystyle C \) is the quarter circle of radius 1 in the xy-plane, from (1, 0, 0) to (0, 1, 0).
- Parametrize: r(t) = <cos(t), sin(t), 0>, 0 ≤ t ≤ pi/2; ds = ‖r′(t)‖ dt.
- \[ \sqrt{\left(\frac{d}{d t} 0\right)^{2} + \left(\frac{d}{d t} \sin{\left(t \right)}\right)^{2} + \left(\frac{d}{d t} \cos{\left(t \right)}\right)^{2}} = 1 \]‖r′(t)‖.✓ Proved
- \[ \int\limits_{0}^{\frac{\pi}{2}} \left. x + y + z \right|_{\substack{ x=\cos{\left(t \right)}\\ y=\sin{\left(t \right)}\\ z=0 }} \cdot 1\, dt = 2 \]∫ f(r(t)) ‖r′(t)‖ dt.✓ Proved
Answer \( 2 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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: fail (error) — The stated answer is 2, but the integral evaluates to 1. The solution claims the integral equals 2, which is mathematically incorrect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer is 2, but the integral evaluates to 1. The solution claims the integral equals 2, which is mathematically incorrect.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The final integral evaluates to 1, not 2. The integral of (cos(t) + sin(t)) from 0 to pi/2 is [sin(t) - cos(t)] from 0 to pi/2, which equals (1 - 0)gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-14 — The final integral evaluates to 2 - sqrt(2), not 2. The solution claims the integral equals 2, which is mathematically incorrect.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 2026-10-18 with SymPy 1.14.0.