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Arc length of parametric curves

Problem 8.130 · medium

Find the length of the curve \( \displaystyle x = 3 \cos{\left(t \right)} \), \( \displaystyle y = 3 \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \frac{2 \pi}{3} \).

\[ \left[\begin{matrix}\frac{d}{d t} 3 \cos{\left(t \right)}\\\frac{d}{d t} 3 \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- 3 \sin{\left(t \right)}\\3 \cos{\left(t \right)}\end{matrix}\right] \]
Velocity components.✓ Proved
\[ 9 \sin^{2}{\left(t \right)} + 9 \cos^{2}{\left(t \right)} = 9 \]
(dx/dt)² + (dy/dt)², simplified.✓ Proved
\[ \int\limits_{0}^{\frac{2 \pi}{3}} 3\, dt = 2 \pi \]
Integrate the speed.✓ Proved

Answer \( 2 \pi \approx 6.28319 \)

✓ 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. The derivatives, simplification of the speed, and definite integral are all correct.

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. The derivatives, simplification of the speed, and definite integral are all correct.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly computes the derivatives, simplifies the speed to a constant 3, and integrates over the given interval to obtain the correct arc length.

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.