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Area with Green's theorem

Problem 12.186 · easy

Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x^{2} y\right) dx + \left(x^{3}\right) dy \), where \( \displaystyle C \) is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), oriented counterclockwise.

\[ \frac{d}{d x} x^{3} - \frac{\partial}{\partial y} x^{2} y = 2 x^{2} \]
∂Q/∂x − ∂P/∂y.✓ Proved
\[ \int\limits_{0}^{3}\int\limits_{0}^{2} 2 x^{2}\, dy\, dx = 36 \]
∬_D (∂Q/∂x − ∂P/∂y) dA.✓ Proved

Answer \( 36 \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.

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 the resulting double integral over the specified rectangular region.

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 the resulting double integral over the specified rectangular region.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies Green's Theorem, computing the partial derivatives and the resulting double integral over the specified rectangular region.
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.