Arc length of x = g(y)
Problem 5.615 · easy
Find the length of the curve \( \displaystyle x = 3 - 3 y \) for \( \displaystyle 0 \le y \le 4 \).
- For x = g(y), L = ∫ √(1 + (dx/dy)²) dy.Reviewed
- \[ \frac{d}{d y} \left(3 - 3 y\right) = -3 \]dx/dy.✓ Proved
- \[ \left(\frac{d}{d y} \left(3 - 3 y\right)\right)^{2} + 1 = 10 \]1 + (dx/dy)², simplified.✓ Proved
- \[ \int\limits_{0}^{4} \sqrt{10}\, dy = 4 \sqrt{10} \]Integrate.✓ Proved
Answer \( 4 \sqrt{10} \)
✓ 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 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.