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Arc length of x = g(y)

Problem 5.691 · hard

Find the length of the curve \( \displaystyle x = \frac{2 y^{\frac{3}{2}}}{3} \) for \( \displaystyle 0 \le y \le 8 \).

For x = g(y), L = ∫ √(1 + (dx/dy)²) dy.
Reviewed
\[ \frac{d}{d y} \frac{2 y^{\frac{3}{2}}}{3} = \sqrt{y} \]
dx/dy.✓ Proved
\[ \left(\frac{d}{d y} \frac{2 y^{\frac{3}{2}}}{3}\right)^{2} + 1 = y + 1 \]
1 + (dx/dy)², simplified.✓ Proved
\[ \int\limits_{0}^{8} \sqrt{y + 1}\, dy = \frac{52}{3} \]
Integrate.✓ Proved

Answer \( \frac{52}{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ 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	a 20,000-segment polyline along the curve

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the arc length formula for a function of y, computes the derivative and its square accurately, and evaluates the definite integral with the correct bounds.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the arc length formula for a function of y, computes the derivative and its square accurately, and evaluates the definite integral with the correct bounds.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the arc length formula for a curve defined as x = g(y). The derivative, simplification, and integration are all correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the arc length formula for a curve defined as x = g(y). The derivative, simplification, and integration are all correct.
gpt-oss:20b: pass 2026-10-16

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/arc_length_y, checked 2026-10-19 with SymPy 1.14.0.