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Arc length in polar coordinates

Problem 8.411 · medium

Find the length of the polar curve \( \displaystyle r = 3 \cos{\left(\theta \right)} \) for \( \displaystyle 0 \le \theta \le \frac{\pi}{2} \).

L = ∫ √(r² + (dr/dθ)²) dθ.
Reviewed
\[ 9 \cos^{2}{\left(\theta \right)} + \left(\frac{d}{d \theta} 3 \cos{\left(\theta \right)}\right)^{2} = 9 \]
r² + (dr/dθ)², simplified.✓ Proved
\[ \int\limits_{0}^{\frac{\pi}{2}} 3\, d\theta = \frac{3 \pi}{2} \]
Integrate.✓ Proved

Answer \( \frac{3 \pi}{2} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the arc length formula for polar curves, simplifies the integrand accurately, and computes the definite integral over the specified interval.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the arc length formula for polar curves, simplifies the integrand accurately, and computes the definite integral over the specified interval.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the polar arc length formula, simplifies the integrand accurately, and computes the definite integral correctly.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-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 correctly.
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.