Arc length in polar coordinates
Problem 8.404 · medium
Find the length of the polar curve \( \displaystyle r = \theta^{2} \) for \( \displaystyle 0 \le \theta \le 1 \).
- L = ∫ √(r² + (dr/dθ)²) dθ.
- \[ \theta^{4} + \left(\frac{d}{d \theta} \theta^{2}\right)^{2} = \theta^{2} \left(\theta^{2} + 4\right) \]r² + (dr/dθ)², simplified.✓ Proved
- \[ \int\limits_{0}^{1} \sqrt{\theta^{2} + 4} \left|{\theta}\right|\, d\theta = - \frac{8}{3} + \frac{5 \sqrt{5}}{3} \]Integrate.✓ Proved
Answer \( - \frac{8}{3} + \frac{5 \sqrt{5}}{3} \)
Lines: 2 proved, 1 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 | ✓ 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: inconclusive — reviewer returned a non-object
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the polar arc length formula, simplifies the integrand, and evaluates the definite integral to match the stated answer.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.