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Area with Green's theorem

Problem 12.249 · easy

Use Green's theorem to find the area enclosed by the ellipse \( \displaystyle \frac{x^2}{9} + \frac{y^2}{4} = 1 \).

Area = ½∮ (x dy − y dx). Parametrize x = a cos t, y = b sin t, 0 ≤ t ≤ 2π.
Reviewed
\[ 6 \sin^{2}{\left(t \right)} + 6 \cos^{2}{\left(t \right)} = 6 \]
x·y′ − y·x′ simplifies to ab.✓ Proved
\[ \frac{\int\limits_{0}^{2 \pi} \left(6 \sin^{2}{\left(t \right)} + 6 \cos^{2}{\left(t \right)}\right)\, dt}{2} = 6 \pi \]
½∫₀^{2π} ab dt.✓ Proved

Answer \( 6 \pi \)

✓ 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	shoelace area of a 20000-gon inscribed in the ellipse (agrees to 1 part in 10⁵)

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies Green's Theorem for area using the standard parametrization of an ellipse. The algebraic simplification and integration are correct, leading to the right answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies Green's Theorem for area using the standard parametrization of an ellipse. The algebraic simplification and integration are correct, leading to the right answer.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies Green's Theorem for area using the formula A = 1/2 ∮(x dy - y dx). The parametrization and subsequent integration are correct, yielding the standard area of an ellipse πab = 6π.

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 2026-10-07 with SymPy 1.14.0.