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Arc length

Problem 5.112 · medium

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

Arc length is ∫ √(1 + (dy/dx)²) dx.
Reviewed
\[ \frac{d}{d x} \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} = \sqrt{x + 2} \]
dy/dx.✓ Proved
\[ x + 3 \]
1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
\[ \int\limits_{1}^{3} \sqrt{x + 3}\, dx = - \frac{16}{3} + 4 \sqrt{6} \]
Integrate.✓ Proved

Answer \( - \frac{16}{3} + 4 \sqrt{6} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the arc length formula, computes the derivative and its square plus one, simplifies the integrand to a perfect square, and evaluates the definite integral correctly.

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 its square plus one, simplifies the integrand to a perfect square, and evaluates the definite integral correctly.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the arc length formula, computes the derivative and simplifies the integrand to a perfect square, and evaluates the definite integral correctly.
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.