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Arc length of parametric curves

Problem 8.126 · medium

Find the length of the curve \( \displaystyle x = 2 t + 1 \), \( \displaystyle y = 5 t - 2 \), \( \displaystyle 0 \le t \le 2 \).

\[ \left[\begin{matrix}\frac{d}{d t} \left(2 t + 1\right)\\\frac{d}{d t} \left(5 t - 2\right)\end{matrix}\right] = \left[\begin{matrix}2\\5\end{matrix}\right] \]
Velocity components.✓ Proved
\[ 29 \]
(dx/dt)² + (dy/dt)², simplified.✓ Proved
\[ \int\limits_{0}^{2} \sqrt{29}\, dt = 2 \sqrt{29} \]
Integrate the speed.✓ Proved

Answer \( 2 \sqrt{29} \approx 10.77033 \)

✓ 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 applies the arc length formula for parametric curves, computes the derivatives and their sum of squares accurately, and evaluates the definite integral correctly.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the arc length formula for parametric curves, computes the derivatives and their sum of squares accurately, and evaluates the definite integral correctly.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06

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.