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Arc length of parametric curves

Problem 8.127 · medium

Find the length of the curve \( \displaystyle x = 3 t^{2} \), \( \displaystyle y = 2 t^{3} \), \( \displaystyle 0 \le t \le 1 \).

\[ \left[\begin{matrix}\frac{d}{d t} 3 t^{2}\\\frac{d}{d t} 2 t^{3}\end{matrix}\right] = \left[\begin{matrix}6 t\\6 t^{2}\end{matrix}\right] \]
Velocity components.✓ Proved
\[ 36 t^{4} + 36 t^{2} = 36 t^{2} \left(t^{2} + 1\right) \]
(dx/dt)² + (dy/dt)², simplified.✓ Proved
\[ \int\limits_{0}^{1} 6 \sqrt{t^{2} + 1} \left|{t}\right|\, dt = -2 + 4 \sqrt{2} \]
Integrate the speed.✓ Proved

Answer \( -2 + 4 \sqrt{2} \approx 3.65685 \)

✓ 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 arc length formula, simplifies the integrand appropriately for the given interval (where t >= 0), and computes 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 identifies the arc length formula, simplifies the integrand appropriately for the given interval (where t >= 0), and computes 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.