Arc length
Problem 5.187 · hard
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 6 \).
- Arc length is ∫ √(1 + (dy/dx)²) dx.
- \[ \frac{d}{d x} \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} = \sqrt{x + 3} \]dy/dx.✓ Proved
- \[ x + 4 \]1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).
- \[ \int\limits_{1}^{6} \sqrt{x + 4}\, dx = - \frac{10 \sqrt{5}}{3} + \frac{20 \sqrt{10}}{3} \]Integrate.✓ Proved
Answer \( - \frac{10 \sqrt{5}}{3} + \frac{20 \sqrt{10}}{3} \)
Lines: 2 proved, 2 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. How we verify
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 | Not checked | — | 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: fail (misleading) — The statement that 1+(dy/dx)² simplifies to a perfect square is incorrect; x+4 is not a perfect square, though the subsequent integration is correct.qwen3.6:27b-mlx: pass — The solution correctly applies the arc length formula, computes the derivative and integrand accurately, and evaluates the definite integral to match the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the arc length formula, computes the derivative and integrand accurately, and evaluates the definite integral to match the stated answer.gpt-oss:20b: fail (misleading) 2026-10-04 — The statement that 1+(dy/dx)² simplifies to a perfect square is incorrect; x+4 is not a perfect square, though the subsequent integration is correct.qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the arc length formula, computes the derivative and the integrand, and evaluates the definite integral with the correct bounds. The logic is sound.gpt-oss:20b: pass 2026-09-29
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.