Arc length of parametric curves
Problem 8.121 · medium
Find the length of the curve \( \displaystyle x = 2 t - 2 \sin{\left(t \right)} \), \( \displaystyle y = 2 - 2 \cos{\left(t \right)} \), \( \displaystyle 0 \le t \le 2 \pi \).
- \[ \left[\begin{matrix}\frac{d}{d t} \left(2 t - 2 \sin{\left(t \right)}\right)\\\frac{d}{d t} \left(2 - 2 \cos{\left(t \right)}\right)\end{matrix}\right] = \left[\begin{matrix}2 - 2 \cos{\left(t \right)}\\2 \sin{\left(t \right)}\end{matrix}\right] \]Velocity components.✓ Proved
- \[ \left(2 - 2 \cos{\left(t \right)}\right)^{2} + 4 \sin^{2}{\left(t \right)} = 16 \sin^{2}{\left(\frac{t}{2} \right)} \](dx/dt)² + (dy/dt)², simplified with 1 − cos t = 2 sin²(t/2).✓ Proved
- \[ \int\limits_{0}^{2 \pi} 4 \sin{\left(\frac{t}{2} \right)}\, dt = 16 \]Integrate the speed (sin(t/2) ≥ 0 on [0, 2π]).✓ Proved
Answer \( 16 \approx 16.00000 \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | numerical quadrature of the speed |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the arc length formula, simplifies the integrand using trigonometric identities, and evaluates the definite integral correctly.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the arc length formula, simplifies the integrand using trigonometric identities, and evaluates the definite integral correctly.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly computes the derivatives, simplifies the speed using trigonometric identities, and integrates over the correct interval. The assumption that sin(t/2) is non-negative on [0, 2pi] is valid.
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/parametric_arc_length, checked 2026-10-06 with SymPy 1.14.0.