Area in polar coordinates
Problem 8.23 · easy
Find the area of the region inside \( \displaystyle r = \sin{\left(\theta \right)} \).
- 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
- \[ \int\limits_{0}^{\pi} \frac{\sin^{2}{\left(\theta \right)}}{2}\, d\theta = \frac{\pi}{4} \]Integrate.✓ Proved
Answer \( \frac{\pi}{4} \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the formula for area in polar coordinates and the correct interval [0, pi] for tracing the circle r = sin(theta) exactly once. The integration is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the formula for area in polar coordinates and the correct interval [0, pi] for tracing the circle r = sin(theta) exactly once. The integration is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the formula for area in polar coordinates and the correct interval [0, pi] for tracing the circle r = sin(theta) once. The integration result is correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the formula for area in polar coordinates and the correct interval [0, pi] for tracing the circle r = sin(theta) exactly once. The integration result 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.