Arc length
Problem 5.114 · medium
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 2 \).
- Arc length is ∫ √(1 + (dy/dx)²) dx.
- \[ \frac{d}{d x} \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} = \sqrt{x + 1} \]dy/dx.✓ Proved
- \[ x + 2 \]1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
- \[ \int\limits_{1}^{2} \sqrt{x + 2}\, dx = \frac{16}{3} - 2 \sqrt{3} \]Integrate.✓ Proved
Answer \( \frac{16}{3} - 2 \sqrt{3} \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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: fail (misleading) — The statement that 1+(dy/dx)² simplifies to a perfect square is incorrect; x+2 is not a perfect square. This could mislead a student about the nature of the integrand.qwen3.6:27b-mlx: pass — The solution correctly identifies the arc length formula, computes the derivative and simplifies the integrand accurately, and evaluates the definite integral with the correct bounds and result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the arc length formula, computes the derivative and simplifies the integrand accurately, and evaluates the definite integral with the correct bounds and result.gpt-oss:20b: fail (misleading) 2026-09-26 — The statement that 1+(dy/dx)² simplifies to a perfect square is incorrect; x+2 is not a perfect square. This could mislead a student about the nature of the integrand.qwen3.6:27b-mlx: pass 2026-09-26 — 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 2026-09-26
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 2026-09-26 with SymPy 1.14.0.