Arc length

Problem 5.47 · hard

Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 6 \).

Arc length is ∫ √(1 + (dy/dx)²) dx.
Reviewed
\[ \frac{d}{d x} \frac{2 x^{\frac{3}{2}}}{3} = \sqrt{x} \]
dy/dx.✓ Proved
\[ x + 1 \]
1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).Reviewed
\[ \int\limits_{1}^{6} \sqrt{x + 1}\, dx = - \frac{4 \sqrt{2}}{3} + \frac{14 \sqrt{7}}{3} \]
Integrate.✓ Proved

Answer \( - \frac{4 \sqrt{2}}{3} + \frac{14 \sqrt{7}}{3} \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the arc length formula, computes the derivative and the simplified integrand, 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 the simplified integrand, and evaluates the definite integral with the correct bounds.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies 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 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the arc length formula, computes the derivative and simplification 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 incorrect; 1+(dy/dx)² = x+1, which is not a perfect square. The rest of the solution is correct.

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.