Arc length of parametric curves
Problem 8.96 · medium
Find the length of the curve \( \displaystyle x = 2 \cos{\left(t \right)} \), \( \displaystyle y = 2 \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \pi \).
- \[ \left[\begin{matrix}\frac{d}{d t} 2 \cos{\left(t \right)}\\\frac{d}{d t} 2 \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- 2 \sin{\left(t \right)}\\2 \cos{\left(t \right)}\end{matrix}\right] \]Velocity components.✓ Proved
- \[ 4 \sin^{2}{\left(t \right)} + 4 \cos^{2}{\left(t \right)} = 4 \](dx/dt)² + (dy/dt)², simplified.✓ Proved
- \[ \int\limits_{0}^{\pi} 2\, dt = 2 \pi \]Integrate the speed.✓ Proved
Answer \( 2 \pi \approx 6.28319 \)
✓ 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.