Arc length
Problem 5.46 · hard
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 2 \) to \( \displaystyle x = 3 \).
- Arc length is ∫ √(1 + (dy/dx)²) dx.Reviewed
- \[ \frac{d}{d x} \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} = \sqrt{x + 3} \]dy/dx.✓ Proved
- \[ x + 4 \]1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).Reviewed
- \[ \int\limits_{2}^{3} \sqrt{x + 4}\, dx = - 4 \sqrt{6} + \frac{14 \sqrt{7}}{3} \]Integrate.✓ Proved
Answer \( - 4 \sqrt{6} + \frac{14 \sqrt{7}}{3} \)
Lines: 2 proved, 2 reviewed. 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. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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 simplifies the integrand accurately, and evaluates the definite integral with the correct bounds.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — 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.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the arc length formula, computes the derivative and simplifies the integrand, and evaluates the definite integral with the correct bounds.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — 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.gpt-oss:20b: fail (misleading) 2026-09-26 — The statement that 1+(dy/dx)² simplifies to a perfect square is false; x+4 is not a perfect square. This could mislead a student into thinking the integrand is a perfect square when it is not.
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 2026-10-13 with SymPy 1.14.0.