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Area in polar coordinates

Problem 8.29 · easy

Find the area of one petal of the rose \( \displaystyle r = 2 \cos{\left(3 \theta \right)} \).

Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
Here θ runs from -pi/6 to pi/6.
\[ \int\limits_{- \frac{\pi}{6}}^{\frac{\pi}{6}} 2 \cos^{2}{\left(3 \theta \right)}\, d\theta = \frac{\pi}{3} \]
Integrate.✓ Proved

Answer \( \frac{\pi}{3} \)

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: fail (error) — The solution omits the factor of 1/2 required by the polar area formula A = 1/2 ∫ r² dθ. The integral computed is ∫ r² dθ, which yields π/3, but the actual area is half of that, π/6.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution omits the factor of 1/2 required by the polar area formula A = 1/2 ∫ r² dθ. The integral computed is ∫ r² dθ, which yields π/3, but the actual area is half of that, π/6.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution omits the factor of 1/2 required by the polar area formula A = 1/2 ∫ r² dθ. The integral computed in step 3 is ∫ r² dθ, so the final answer should be half of pi/3, which is pi/6.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution omits the factor of 1/2 required by the polar area formula A = 1/2 ∫ r² dθ. Consequently, the calculated value is double the correct area.
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.