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Area in polar coordinates

Problem 8.30 · easy

Find the area of the region inside \( \displaystyle r = 2 \sin{\left(\theta \right)} \).

Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
Here θ runs from 0 to pi.
\[ \int\limits_{0}^{\pi} 2 \sin^{2}{\left(\theta \right)}\, d\theta = \pi \]
Integrate.✓ Proved

Answer \( \pi \)

Lines: 1 proved, 2 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	Not checked	—	a sentence; read, not computed
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 shoelace formula on 20,000 points of the curve gives the same area

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution states the formula for polar area involves a factor of 1/2, but the integral in step 3 omits this factor, effectively calculating 2 * A

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution states the formula for polar area involves a factor of 1/2, but the integral in step 3 omits this factor, effectively calculating 2 * A
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution states the formula involves 1/2, but the integral computed in step 3 is missing the 1/2 factor. The integral of 2*sin^2(theta) from 0 to pi is pi, so the area should be pi/2, not pi.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution states the formula for area is 1/2 * integral(r^2 dtheta), but the computed integral in step 3 is missing the factor of 1/2. The integral of 2*sin^2(theta) from 0 to pi is pi, so the area should be pi/2, not pi.
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/polar_area, checked 2026-09-28 with SymPy 1.14.0.