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Area with Green's theorem

Problem 12.275 · easy

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

\[ - \frac{d}{d y} 0 + \frac{d}{d x} 2 x = 2 \]
∂Q/∂x − ∂P/∂y.✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{3} 2\, dy\, dx = 12 \]
∬_D (∂Q/∂x − ∂P/∂y) dA.✓ Proved

Answer \( 12 \)

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 identifies the integrand for Green's Theorem and computes the double integral over the specified rectangular region. The steps are logically sound and the result is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the integrand for Green's Theorem and computes the double integral over the specified rectangular region. The steps are logically sound and the result is correct.
qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-object
gpt-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.