Arc length in polar coordinates
Problem 8.283 · medium
Find the length of the polar curve \( \displaystyle r = e^{\theta} \) for \( \displaystyle 0 \le \theta \le 1 \).
- L = ∫ √(r² + (dr/dθ)²) dθ.Reviewed
- \[ e^{2 \theta} + \left(\frac{d}{d \theta} e^{\theta}\right)^{2} = 2 e^{2 \theta} \]r² + (dr/dθ)², simplified.✓ Proved
- \[ \int\limits_{0}^{1} \sqrt{2} e^{\theta}\, d\theta = \sqrt{2} \left(-1 + e\right) \]Integrate.✓ Proved
Answer \( \sqrt{2} \left(-1 + e\right) \)
✓ 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 arc length formula for polar curves, simplifies the integrand accurately, and evaluates the definite integral with the correct bounds.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the arc length formula for polar curves, simplifies the integrand accurately, and evaluates the definite integral with the correct bounds.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 accurately, and computes the definite integral with the correct bounds.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, simplifies the integrand accurately, and computes the definite integral with the correct bounds.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.