Area with Green's theorem
Problem 12.212 · easy
Use Green's theorem to find the area enclosed by the ellipse \( \displaystyle \frac{x^2}{4} + \frac{y^2}{4} = 1 \).
- Area = ½∮ (x dy − y dx). Parametrize x = a cos t, y = b sin t, 0 ≤ t ≤ 2π.
- \[ 4 \sin^{2}{\left(t \right)} + 4 \cos^{2}{\left(t \right)} = 4 \]x·y′ − y·x′ simplifies to ab.✓ Proved
- \[ \frac{\int\limits_{0}^{2 \pi} \left(4 \sin^{2}{\left(t \right)} + 4 \cos^{2}{\left(t \right)}\right)\, dt}{2} = 4 \pi \]½∫₀^{2π} ab dt.✓ Proved
Answer \( 4 \pi \)
Lines: 2 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The solution omits the explicit values a=2 and b=2 for the ellipse. Without stating that the ellipse has semi‑axes 2, the parametrization and the simplification of x y′−y x′ to ab are incomplete, leading to an incorrect or incomplete setup.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The parametrization uses a=2, b=2, so ab=4. The integral of 4 from 0 to 2pi is 8pi. Dividing by 2 gives 4pi. Wait, the area of a circle with radius 2
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The solution omits the explicit values a=2 and b=2 for the ellipse. Without stating that the ellipse has semi‑axes 2, the parametrization and the simplification of x y′−y x′ to ab are incomplete, leading to an incorrect or incomplete setup.qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The parametrization uses a=2, b=2, so ab=4. The integral of 4 from 0 to 2pi is 8pi. Dividing by 2 gives 4pi. Wait, the area of a circle with radius 2gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the curve as a circle of radius 2 (since a=2, b=2) and applies the standard area formula derived from Green's theorem. The algebraic simplification and integration are correct.
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 2026-10-06 with SymPy 1.14.0.