Area with Green's theorem
Problem 12.279 · medium
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 \( \displaystyle x^2 + y^2 = 9 \), oriented counterclockwise.
- \[ \frac{d}{d x} x^{2} - \frac{\partial}{\partial y} x^{2} y = - x^{2} + 2 x \]∂Q/∂x − ∂P/∂y.✓ Proved
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3} r \left(- r^{2} \cos^{2}{\left(\theta \right)} + 2 r \cos{\left(\theta \right)}\right)\, dr\, d\theta = - \frac{81 \pi}{4} \]∬_D (∂Q/∂x − ∂P/∂y) dA in polar coordinates.✓ Proved
Answer \( - \frac{81 \pi}{4} \)
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: pass — The solution correctly applies Green's Theorem, computes the partial derivatives accurately, and sets up the polar integral with the correct bounds and Jacobian. The final result is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies Green's Theorem, computes the partial derivatives accurately, and sets up the polar integral with the correct bounds and Jacobian. The final result is correct.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The partial derivatives are incorrect: ∂Q/∂x should be 2x and ∂P/∂y should be x², so the integrand is 2x - x², not -x² + 2x (which implies ∂Q/∂x = 2xgpt-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.