Arc length

Problem 5.56 · hard

Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 3\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 4 \).

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
\[ \int\limits_{1}^{4} \sqrt{x + 4}\, dx = - \frac{10 \sqrt{5}}{3} + \frac{32 \sqrt{2}}{3} \]
Integrate.✓ Proved

Answer \( - \frac{10 \sqrt{5}}{3} + \frac{32 \sqrt{2}}{3} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 simplifies the integrand, and evaluates the definite integral with the correct bounds and result.

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, and evaluates the definite integral with the correct bounds and result.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the arc length formula, computes the derivative and the term inside the square root, and evaluates the definite integral with the correct bounds. The logic is sound and the result matches the stated answer.
gpt-oss:20b: pass 2026-09-27
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: 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 2026-10-13 with SymPy 1.14.0.