Arc length of parametric curves
Problem 8.153 · medium
Find the length of the curve \( \displaystyle x = 5 t + 1 \), \( \displaystyle y = - 2 t - 2 \), \( \displaystyle 0 \le t \le 4 \).
- \[ \left[\begin{matrix}\frac{d}{d t} \left(5 t + 1\right)\\\frac{d}{d t} \left(- 2 t - 2\right)\end{matrix}\right] = \left[\begin{matrix}5\\-2\end{matrix}\right] \]Velocity components.✓ Proved
- \[ 29 \](dx/dt)² + (dy/dt)², simplified.✓ Proved
- \[ \int\limits_{0}^{4} \sqrt{29}\, dt = 4 \sqrt{29} \]Integrate the speed.✓ Proved
Answer \( 4 \sqrt{29} \approx 21.54066 \)
✓ 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 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the arc length formula for parametric curves. The derivatives, sum of squares, and definite integral are all correct.gpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.