Home›Calculus 2›Area in polar coordinates›Problem 8.22

Area in polar coordinates

Problem 8.22 · easy

Find the area of one petal of the rose \( \displaystyle r = 4 \cos{\left(2 \theta \right)} \).

Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
Reviewed
Here θ runs from -pi/4 to pi/4.
Reviewed
\[ \int\limits_{- \frac{\pi}{4}}^{\frac{\pi}{4}} 8 \cos^{2}{\left(2 \theta \right)}\, d\theta = 2 \pi \]
Integrate.✓ Proved

Answer \( 2 \pi \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the bounds for one petal of the rose curve r = 4 cos(2θ) and applies the polar area formula accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the bounds for one petal of the rose curve r = 4 cos(2θ) and applies the polar area formula accurately.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The setup correctly identifies the bounds for one petal of the rose curve r = 4 cos(2θ) as [-π/4, π/4]. The integration and final result are correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the bounds for one petal of the rose curve r = 4 cos(2θ) and applies the standard polar area formula. The calculation is correct.
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.