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Arc length in polar coordinates

Problem 8.289 · medium

Find the length of the polar curve \( \displaystyle r = 4 \cos{\left(\theta \right)} + 4 \) for \( \displaystyle 0 \le \theta \le \pi \).

L = ∫ √(r² + (dr/dθ)²) dθ.
Reviewed
\[ \left(4 \cos{\left(\theta \right)} + 4\right)^{2} + \left(\frac{d}{d \theta} \left(4 \cos{\left(\theta \right)} + 4\right)\right)^{2} = 32 \cos{\left(\theta \right)} + 32 \]
r² + (dr/dθ)², simplified.✓ Proved
\[ 64 \cos^{2}{\left(\frac{\theta}{2} \right)} = 32 \cos{\left(\theta \right)} + 32 \]
Its square root is 2a cos(θ/2) on [0, π] (half-angle identity).✓ Proved
\[ \int\limits_{0}^{\pi} 8 \cos{\left(\frac{\theta}{2} \right)}\, d\theta = 16 \]
Integrate.✓ Proved

Answer \( 16 \)

✓ 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, simplifies the integrand using trigonometric identities, and evaluates the definite integral correctly.

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 using trigonometric identities, and evaluates the definite integral correctly.
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. The simplification of the integrand using half-angle identities is valid, and the integration over the specified bounds yields the correct result.
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, simplifies the integrand using half-angle identities, and evaluates the definite integral correctly.
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.