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Arc length of parametric curves

Problem 8.128 · medium

Find the length of the curve \( \displaystyle x = 2 t + 1 \), \( \displaystyle y = - 4 t - 2 \), \( \displaystyle 0 \le t \le 1 \).

\[ \left[\begin{matrix}\frac{d}{d t} \left(2 t + 1\right)\\\frac{d}{d t} \left(- 4 t - 2\right)\end{matrix}\right] = \left[\begin{matrix}2\\-4\end{matrix}\right] \]
Velocity components.✓ Proved
\[ 20 \]
(dx/dt)² + (dy/dt)², simplified.✓ Proved
\[ \int\limits_{0}^{1} 2 \sqrt{5}\, dt = 2 \sqrt{5} \]
Integrate the speed.✓ Proved

Answer \( 2 \sqrt{5} \approx 4.47214 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the velocity vector, computes the speed as a constant, and integrates it over the given interval to find the arc length.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the velocity vector, computes the speed as a constant, and integrates it over the given interval to find the arc length.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the velocity vector, computes the speed as a constant, and integrates it over the given interval. The steps are logically sound and the final answer is correct.

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.