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Area in polar coordinates

Problem 8.46 · easy

Find the area of the region inside \( \displaystyle r = 4 \sin{\left(\theta \right)} \).

Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
Reviewed
Here θ runs from 0 to pi.
Reviewed
\[ \int\limits_{0}^{\pi} 8 \sin^{2}{\left(\theta \right)}\, d\theta = 4 \pi \]
Integrate.✓ Proved

Answer \( 4 \pi \)

✓ 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 formula for area in polar coordinates, determines the correct interval [0, π] for tracing the circle r = 4 sin(θ) exactly once, and computes the integral correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the formula for area in polar coordinates, determines the correct interval [0, π] for tracing the circle r = 4 sin(θ) exactly once, and computes the integral correctly.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the bounds for a single traversal of the circle r = 4 sin(theta) and applies the polar area formula accurately.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the bounds for a single traversal of the circle r = 4 sin(theta) and applies the polar area formula accurately.
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.