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Area with Green's theorem

Problem 12.180 · easy

Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x y\right) dx + \left(2 x\right) dy \), where \( \displaystyle C \) is the triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (1, 0) \), \( \displaystyle (0, 1) \), oriented counterclockwise.

\[ \frac{d}{d x} 2 x - \frac{\partial}{\partial y} x y = 2 - x \]
∂Q/∂x − ∂P/∂y.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{1 - x} \left(2 - x\right)\, dy\, dx = \frac{5}{6} \]
∬_D (∂Q/∂x − ∂P/∂y) dA.✓ Proved

Answer \( \frac{5}{6} \)

Every line of this solution was proved by the computer algebra system SymPy. 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.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ 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	the line integral around the boundary, computed directly

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies Green's Theorem, computing the partial derivatives and setting up the iterated integral over the triangular region with correct bounds and orientation.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies Green's Theorem, computing the partial derivatives and setting up the iterated integral over the triangular region with correct bounds and orientation.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution calculates the double integral of (2 - x) over the region, but the problem asks to use Green's Theorem, which requires computing the double integral of (∂Q/∂x - ∂P/∂y). Here P = xy and Q = 2x, so ∂Q/∂x = 2 and ∂P/∂y = x. The integrand should be (2 - x). Wait, the solution says 'Derivative(2*x, x) - Derivative(x*y, y) = 2 - x'. This is correct for the integrand. However, the final answer 5/6 is incorrect for this specific integral over the triangle. Let's re-evaluate: Integral from x=0 to 1 of Integral from y=0 to 1-x of (2-x) dy dx. Inner integral: (2-x)[y] from 0 to 1-x = (2-x)(1-x). Outer integral: Integral from 0 to 1 of (2 - 3x + x^2) dx = [2x - 3/2 x^2 + 1/3 x^3] from 0 to 1 = 2 - 1.5 + 1/3 = 0.5 + 0.333 = 5/6. The calculation is actually correct. Why did I think it was wrong? Let me re-read the prompt. 'Stated answer: 5/6'. The solution derives 5/6. Is there a sign error? Green's theorem: ∮ P dx + Q dy = ∬ (∂Q/∂x - ∂P/∂y) dA. P=xy, Q=2x. ∂Q/∂x = 2. ∂P/∂y = x. Integrand 2-x. Region is triangle (0,0), (1,0), (0,1). Orientation is counterclockwise. The setup is correct. The calculation is correct. The answer is correct. Wait, is there a subtle issue? The prompt says 'Review it'. If it's correct, I should say pass. Let me double check the integral again. ∫₀¹ (2-x)(1-x) dx = ∫₀¹ (2 - 3x + x²) dx = [2x - 3/2 x² + 1/3 x³]₀¹ = 2 - 3/2 + 1/3 = 12/6 - 9/6 + 2/6 = 5/6. The solution is correct. Why would it fail? Maybe the 'unchecked' tag implies I shouldn't check the algebra? No, the instructions say 'check that... each conclusion follows from the lines before it'. The lines are equations marked unchecked, but the text says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. Wait, the lines are marked '[equation, unchecked]'. The instructions say 'Each line... is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. The input format shows '[equation, unchecked]'. This might mean the CAS check was skipped or failed? Or does 'unchecked' mean I *should* check it? The prompt says 'do not re-check the algebra of a line marked proved'. These are marked 'unchecked'. This is ambiguous. However, usually in these tasks, if the math is right, it's a pass. Let's look for a 'sentence' error. There are no sentences. Just two equations. The prompt asks to review 'sentences'. If there are no sentences, is it a pass? Or is the lack of explanation a style issue? The prompt says 'The sentences are what you are here for.' If there are no sentences, maybe I should flag it for lack of setup? But the equations implicitly show the setup. Let's look closer. Is the orientation correct? Counterclockwise. Green's theorem applies directly. Is the region correct? Triangle vertices (0,0), (1,0), (0,1). Yes. Is the integrand correct? Yes. Is the integral evaluation correct? Yes. I will mark it as pass. Wait, looking at similar examples, sometimes 'unchecked' means the user didn't run the CAS, so I might need to verify. I verified it, it's correct. So pass.
gpt-oss:20b: pass 2026-10-04

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/greens_area, checked 2026-10-04 with SymPy 1.14.0.