Arc length

Problem 5.58 · medium

Find the length of the curve \( \displaystyle y = \frac{x^{4}}{8} + \frac{1}{4 x^{2}} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 2 \).

Arc length is ∫ √(1 + (dy/dx)²) dx.
Reviewed
\[ \frac{d}{d x} \left(\frac{x^{4}}{8} + \frac{1}{4 x^{2}}\right) = \frac{x^{3}}{2} - \frac{1}{2 x^{3}} \]
dy/dx.✓ Proved
\[ \left(\frac{x^{3}}{2} - \frac{1}{2 x^{3}}\right)^{2} + 1 = \frac{\left(x^{2} + 1\right)^{2} \left(x^{4} - x^{2} + 1\right)^{2}}{4 x^{6}} \]
1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
\[ \int\limits_{1}^{2} \sqrt{\left(\frac{x^{3}}{2} - \frac{1}{2 x^{3}}\right)^{2} + 1}\, dx = \frac{33}{16} \]
Integrate.✓ Proved

Answer \( \frac{33}{16} \)

✓ 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. 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	✓ 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 applies the arc length formula, computes the derivative, simplifies the integrand (noting the perfect square structure typical of such problems), and evaluates the definite integral to the correct result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the arc length formula, computes the derivative, simplifies the integrand (noting the perfect square structure typical of such problems), and evaluates the definite integral to the correct result.
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 evaluates the definite integral to the correct result. The simplification in step 3 is algebraically correct, although the expression shown is not the simplest perfect square form (which would be ((x^3+1)/(2x^3))^2 or similar), but it does not affect the final correctness.
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 evaluates the integral. Although the intermediate simplification in line 3 is unnecessarily complex and does not explicitly show the perfect square structure that makes the square root trivial, the final result is correct and the logic holds.
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 2026-09-28 with SymPy 1.14.0.