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Arc length in polar coordinates

Problem 8.294 · medium

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

L = ∫ √(r² + (dr/dθ)²) dθ.
Reviewed
\[ \left(3 \cos{\left(\theta \right)} + 3\right)^{2} + \left(\frac{d}{d \theta} \left(3 \cos{\left(\theta \right)} + 3\right)\right)^{2} = 18 \cos{\left(\theta \right)} + 18 \]
r² + (dr/dθ)², simplified.✓ Proved
\[ 36 \cos^{2}{\left(\frac{\theta}{2} \right)} = 18 \cos{\left(\theta \right)} + 18 \]
Its square root is 2a cos(θ/2) on [0, π] (half-angle identity).✓ Proved
\[ \int\limits_{0}^{\pi} 6 \cos{\left(\frac{\theta}{2} \right)}\, d\theta = 12 \]
Integrate.✓ Proved

Answer \( 12 \)

✓ Nihil obstat Lines: 3 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
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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the arc length formula for polar curves. The simplification of the integrand using half-angle identities is valid, and the integration limits and result are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the arc length formula for polar curves. The simplification of the integrand using half-angle identities is valid, and the integration limits and result are correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — 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: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the arc length formula for polar curves. The simplification of the integrand using the half-angle identity is valid, and the integration bounds and result are correct.
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.