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Arc length in polar coordinates

Problem 8.414 · medium

Find the length of the polar curve \( \displaystyle r = \cos{\left(\theta \right)} + 1 \) for \( \displaystyle 0 \le \theta \le \pi \).

L = ∫ √(r² + (dr/dθ)²) dθ.
\[ \left(\cos{\left(\theta \right)} + 1\right)^{2} + \left(\frac{d}{d \theta} \left(\cos{\left(\theta \right)} + 1\right)\right)^{2} = 2 \cos{\left(\theta \right)} + 2 \]
r² + (dr/dθ)², simplified.✓ Proved
\[ 4 \cos^{2}{\left(\frac{\theta}{2} \right)} = 2 \cos{\left(\theta \right)} + 2 \]
Its square root is 2a cos(θ/2) on [0, π] (half-angle identity).✓ Proved
\[ \int\limits_{0}^{\pi} 2 \cos{\left(\frac{\theta}{2} \right)}\, d\theta = 4 \]
Integrate.✓ Proved

Answer \( 4 \)

Lines: 3 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
4	✓ 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: fail (error) — The step equating √(2cosθ+2) with 2cos(θ/2) is incorrect; the correct simplification is √(2)·cos(θ/2). Consequently the integral and the stated answer 4 are wrong.
qwen3.6:27b-mlx: pass — The solution correctly applies the arc length formula for polar curves, simplifies the integrand using trigonometric identities, and handles the absolute value of the square root correctly over the given 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 using trigonometric identities, and handles the absolute value of the square root correctly over the given interval.
gpt-oss:20b: fail (error) 2026-10-18 — The step equating √(2cosθ+2) with 2cos(θ/2) is incorrect; the correct simplification is √(2)·cos(θ/2). Consequently the integral and the stated answer 4 are wrong.
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the arc length formula for polar curves, simplifies the integrand using half-angle identities, and handles the absolute value of the square root correctly over the given interval.
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. The algebraic simplification using the half-angle identity is valid, and the integration is correct.
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.