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Arc length of parametric curves

Problem 8.98 · medium

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

\[ \left[\begin{matrix}\frac{d}{d t} \left(2 t + 1\right)\\\frac{d}{d t} \left(2 t - 2\right)\end{matrix}\right] = \left[\begin{matrix}2\\2\end{matrix}\right] \]
Velocity components.✓ Proved
\[ 8 \]
(dx/dt)² + (dy/dt)², simplified.✓ Proved
\[ \int\limits_{0}^{4} 2 \sqrt{2}\, dt = 8 \sqrt{2} \]
Integrate the speed.✓ Proved

Answer \( 8 \sqrt{2} \approx 11.31371 \)

✓ 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 speed accurately, and integrates over the correct interval.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the arc length formula for parametric curves, computes the derivatives and speed accurately, and integrates over the correct interval.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the derivatives, simplifies the integrand to a constant, and evaluates the definite integral over the correct bounds.
gpt-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.