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Green's theorem

Problem 12.91 · easy

Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.

Green's theorem: ∮ P dx + Q dy = ∬ (∂Q/∂x − ∂P/∂y) dA.
Reviewed
\[ \frac{d}{d x} x^{2} - \frac{\partial}{\partial y} x y = x \]
∂Q/∂x − ∂P/∂y.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{2} x\, dx\, dy = 2 \]
Integrate over the rectangle.✓ Proved

Answer \( 2 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	the line integral around the four edges, done directly, gives the same value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies Green's theorem, computes the partial derivatives accurately, and evaluates the double integral over the specified rectangular region.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies Green's theorem, computes the partial derivatives accurately, and evaluates the double integral over the specified rectangular region.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies Green's theorem, computes the partial derivatives accurately, and sets up the double integral over the specified rectangular region with correct bounds.
gpt-oss:20b: pass 2026-09-26

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_theorem, checked 2026-09-26 with SymPy 1.14.0.