Arc length of x = g(y)
Problem 5.630 · hard
Find the length of the curve \( \displaystyle x = \frac{2 \left(y - 1\right)^{\frac{3}{2}}}{3} \) for \( \displaystyle 4 \le y \le 9 \).
- For x = g(y), L = ∫ √(1 + (dx/dy)²) dy.Reviewed
- \[ \frac{d}{d y} \frac{2 \left(y - 1\right)^{\frac{3}{2}}}{3} = \sqrt{y - 1} \]dx/dy.✓ Proved
- \[ \left(\frac{d}{d y} \frac{2 \left(y - 1\right)^{\frac{3}{2}}}{3}\right)^{2} + 1 = y \]1 + (dx/dy)², simplified.✓ Proved
- \[ \int\limits_{4}^{9} \sqrt{y}\, dy = \frac{38}{3} \]Integrate.✓ Proved
Answer \( \frac{38}{3} \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the arc length formula for a curve defined as x = g(y), computes the derivative and integrand accurately, and evaluates the definite integral with the correct bounds.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the arc length formula for a curve defined as x = g(y), computes the derivative and integrand accurately, and evaluates the definite integral with the correct bounds.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the arc length formula for a curve defined as x = g(y), computes the derivative and integrand accurately, and evaluates the definite integral over the specified bounds.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the arc length formula for x = g(y), computes the derivative and integrand accurately, and evaluates the definite integral with the correct bounds.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.