Arc length in polar coordinates
Problem 8.282 · medium
Find the length of the polar curve \( \displaystyle r = \theta^{2} \) for \( \displaystyle 0 \le \theta \le \sqrt{5} \).
- L = ∫ √(r² + (dr/dθ)²) dθ.Reviewed
- \[ \theta^{4} + \left(\frac{d}{d \theta} \theta^{2}\right)^{2} = \theta^{2} \left(\theta^{2} + 4\right) \]r² + (dr/dθ)², simplified.✓ Proved
- \[ \int\limits_{0}^{\sqrt{5}} \sqrt{\theta^{2} + 4} \left|{\theta}\right|\, d\theta = \frac{19}{3} \]Integrate.✓ Proved
Answer \( \frac{19}{3} \)
✓ Nihil obstat Lines: 2 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | a 20,000-segment polyline along the curve |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the polar arc length formula, simplifies the integrand, and handles the absolute value correctly given the non-negative bounds. The final result is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the polar arc length formula, simplifies the integrand, and handles the absolute value correctly given the non-negative bounds. The final result is correct.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the arc length formula for polar curves, simplifies the integrand, and evaluates the definite integral accurately.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the arc length formula for polar curves. The simplification of the integrand is algebraically correct, and the integration steps (handling the absolute value correctly over the positive domain) yield the stated result.gpt-oss:20b: pass 2026-10-14
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_arc_length, checked 2026-10-17 with SymPy 1.14.0.