Arc length of x = g(y)
Problem 5.718 · easy
Find the length of the curve \( \displaystyle x = 4 y + 2 \) for \( \displaystyle 0 \le y \le 5 \).
- For x = g(y), L = ∫ √(1 + (dx/dy)²) dy.Reviewed
- \[ \frac{d}{d y} \left(4 y + 2\right) = 4 \]dx/dy.✓ Proved
- \[ \left(\frac{d}{d y} \left(4 y + 2\right)\right)^{2} + 1 = 17 \]1 + (dx/dy)², simplified.✓ Proved
- \[ \int\limits_{0}^{5} \sqrt{17}\, dy = 5 \sqrt{17} \]Integrate.✓ Proved
Answer \( 5 \sqrt{17} \)
✓ 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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-23gpt-oss:20b: pass 2026-10-23qwen3.6:27b-mlx: pass 2026-10-22gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly applies the arc length formula for a function of y, computes the derivative and integral accurately, and uses the correct bounds.gpt-oss:20b: pass 2026-10-22
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 2026-10-23 with SymPy 1.14.0.