Line integrals of scalar functions
Problem 12.406 · medium
Evaluate \( \displaystyle \int_C x + y \, ds \), 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
- \[ \int\limits_{0}^{\frac{\pi}{2}} \left. x + y \right|_{\substack{ x=\cos{\left(t \right)}\\ y=\sin{\left(t \right)}\\ z=0 }} \cdot 1\, dt = 2 \]∫ 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 parametrization r(t) = <cos(t), sin(t), 0> traces the curve from (1, 0) to (0, 1) as t goes from 0 to pi/2, which is correct. However, 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) - (0 - 1) = 2. Wait, the calculation is actually correct. Let me re-read the prompt constraints. 'Your job is to find what is wrong'. Is there a subtle error? The problem asks for integral of x+y ds. ds = dt. Integral is 2. The solution says 2. The steps are correct. Why would it fail? Ah, looking closely at step 3: `Integral(Subs(x + y, (x, y, z), (cos(t), sin(t), 0))*1, (t, 0, pi/2)) = 2`. The substitution is correct. The bounds are correct. The result is correct. Is it possible the 'error' is in the sentence description? 'Parametrize: r(t) = <cos(t), sin(t), 0>, 0 ≤ t ≤ pi/2; ds = ‖r′(t)‖ dt.' This is standard. Maybe the issue is that the problem statement says 'from (1, 0, 0) to (0, 1, 0)' and the parametrization does exactly that. Let's look for a 'misleading' or 'style' issue if not an error. Or perhaps I am missing a sign? No, scalar line integrals are independent of orientation. Wait, let's look at the JSON output format required. If it is sound, I should say pass. Is it sound? Yes. Let me double check the 'adversarial' nature. Is there a trap? Problem: Evaluate \int_C x + y ds. C is quarter circle radius 1. Parametrization: x=cos t, y=sin t. ds = sqrt((-sin t)^2 + (cos t)^2) dt = 1 dt. Integral_0^{pi/2} (cos t + sin t) dt = [sin t - cos t]_0^{pi/2} = (1 - 0) - (0 - 1) = 2. The solution is correct. However, often in these adversarial tasks, if the solution is correct, the answer is 'pass'. Let's check if the sentence in step 1 is technically a 'sentence' or if it's just a list of facts. It sets up the model. Is there any ambiguity about 'quarter circle'? Usually implies the first quadrant. The points (1,0) and (0,1) confirm this. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The parametrization r(t) = <cos(t), sin(t), 0> traces the curve from (1, 0) to (0, 1) as t goes from 0 to pi/2, which is correct. However, 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) - (0 - 1) = 2. Wait, the calculation is actually correct. Let me re-read the prompt constraints. 'Your job is to find what is wrong'. Is there a subtle error? The problem asks for integral of x+y ds. ds = dt. Integral is 2. The solution says 2. The steps are correct. Why would it fail? Ah, looking closely at step 3: `Integral(Subs(x + y, (x, y, z), (cos(t), sin(t), 0))*1, (t, 0, pi/2)) = 2`. The substitution is correct. The bounds are correct. The result is correct. Is it possible the 'error' is in the sentence description? 'Parametrize: r(t) = <cos(t), sin(t), 0>, 0 ≤ t ≤ pi/2; ds = ‖r′(t)‖ dt.' This is standard. Maybe the issue is that the problem statement says 'from (1, 0, 0) to (0, 1, 0)' and the parametrization does exactly that. Let's look for a 'misleading' or 'style' issue if not an error. Or perhaps I am missing a sign? No, scalar line integrals are independent of orientation. Wait, let's look at the JSON output format required. If it is sound, I should say pass. Is it sound? Yes. Let me double check the 'adversarial' nature. Is there a trap? Problem: Evaluate \int_C x + y ds. C is quarter circle radius 1. Parametrization: x=cos t, y=sin t. ds = sqrt((-sin t)^2 + (cos t)^2) dt = 1 dt. Integral_0^{pi/2} (cos t + sin t) dt = [sin t - cos t]_0^{pi/2} = (1 - 0) - (0 - 1) = 2. The solution is correct. However, often in these adversarial tasks, if the solution is correct, the answer is 'pass'. Let's check if the sentence in step 1 is technically a 'sentence' or if it's just a list of facts. It sets up the model. Is there any ambiguity about 'quarter circle'? Usually implies the first quadrant. The points (1,0) and (0,1) confirm this. I will mark it as pass.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The parametrization r(t) = <cos(t), sin(t), 0> for 0 ≤ t ≤ pi/2 traces the quarter circle from (1, 0) to (0, 1) in a counter-clockwise direction, which matches the problem statement. However, the integral of x + y over this path is ∫(cos(t) + sin(t)) dt from 0 to pi/2, which equals [sin(t) - cos(t)] from 0 to pi/2 = (1 - 0) - (0 - 1) = 2. Wait, the calculation is correct. Let me re-read the prompt constraints. 'Find what is wrong'. Is there a subtle issue? The problem asks for ∫_C x + y ds. ds is scalar arc length. The direction of traversal does not affect the value of a scalar line integral. The parametrization is correct. The bounds are correct. The integrand substitution is correct. The result 2 is correct. Why would this fail? Perhaps the 'sentence' in step 1 is considered incomplete because it doesn't explicitly state that the direction is correct or that ds is positive? No, that's standard. Let's look closer at the equation line. 'Derivative(0, t)**2 + Derivative(sin(t), t)**2 + Derivative(cos(t), t)**2'. Derivative(cos(t), t) is -sin(t). Squared is sin^2(t). Derivative(sin(t), t) is cos(t). Squared is cos^2(t). Sum is 1. Sqrt is 1. This is correct. The integral is ∫_0^{pi/2} (cos(t) + sin(t)) * 1 dt = 2. This is correct. Is there a 'style' issue? The prompt says 'reply with JSON only'. If it's correct, verdict is pass. But usually these adversarial reviews have a trick. Let's check the start and end points. (1, 0, 0) to (0, 1, 0). t=0 -> (1, 0, 0). t=pi/2 -> (0, 1, 0). Correct. Radius 1. Correct. Quarter circle. Correct. Maybe the issue is that the solution doesn't explicitly write out the integral evaluation steps, just jumps to the answer? The prompt says 'each line... is either an EQUATION... or a SENTENCE'. Line 3 is an equation marked 'proved'. If the CAS proved it, it's correct. Is there any ambiguity in 'quarter circle'? No. Is there any ambiguity in 'from ... to ...' for a scalar integral? No, scalar line integrals are independent of orientation. If it were a vector line integral ∫ F · dr, orientation would matter. But it's ∫ f ds. So orientation doesn't matter. The solution is sound. I will mark it as pass.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer is 2, but the integral evaluates to sqrt(2) - 1. The solution claims the integral equals 2 without showing the calculation, and the value is 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.