Arc length
Problem 5.139 · hard
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 5 \).
- Arc length is ∫ √(1 + (dy/dx)²) dx.Reviewed
- \[ \frac{d}{d x} \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} = \sqrt{x + 1} \]dy/dx.✓ Proved
- \[ x + 2 \]1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
- \[ \int\limits_{0}^{5} \sqrt{x + 2}\, dx = - \frac{4 \sqrt{2}}{3} + \frac{14 \sqrt{7}}{3} \]Integrate.✓ Proved
Answer \( - \frac{4 \sqrt{2}}{3} + \frac{14 \sqrt{7}}{3} \)
✓ 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.
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 200,000-segment polygon along the curve has the same length |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the arc length formula, computes the derivative and its square, simplifies the integrand, and evaluates the definite integral with the correct bounds.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the arc length formula, computes the derivative and its square, simplifies the integrand, and evaluates the definite integral with the correct bounds.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the arc length formula, computes the derivative and the simplified integrand, and evaluates the definite integral with the correct bounds and result.gpt-oss:20b: pass 2026-09-27
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, checked 2026-09-27 with SymPy 1.14.0.