Area with Green's theorem
Problem 12.277 · easy
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x^{2} y\right) dx + \left(x^{2}\right) dy \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 1 \), oriented counterclockwise.
- \[ \frac{d}{d x} x^{2} - \frac{\partial}{\partial y} x^{2} y = - x^{2} + 2 x \]∂Q/∂x − ∂P/∂y.✓ Proved
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1} r \left(- r^{2} \cos^{2}{\left(\theta \right)} + 2 r \cos{\left(\theta \right)}\right)\, dr\, d\theta = - \frac{\pi}{4} \]∬_D (∂Q/∂x − ∂P/∂y) dA in polar coordinates.✓ Proved
Answer \( - \frac{\pi}{4} \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The partial derivatives are calculated incorrectly: ∂Q/∂x should be 2x and ∂P/∂y should be x², so ∂Q/∂x - ∂P/∂y = 2x - x², not -x² + 2x (which implie
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The partial derivatives are calculated incorrectly: ∂Q/∂x should be 2x and ∂P/∂y should be x², so ∂Q/∂x - ∂P/∂y = 2x - x², not -x² + 2x (which implieqwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The integrand for the double integral is incorrect. The term 2*r*cos(theta) comes from differentiating x^2*y with respect to y, but the problem defingpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.