Arc length

Problem 5.60 · medium

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

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
\[ \int\limits_{1}^{4} \sqrt{\left(\frac{x^{3}}{2} - \frac{1}{2 x^{3}}\right)^{2} + 1}\, dx = \frac{2055}{64} \]
Integrate.✓ Proved

Answer \( \frac{2055}{64} \)

Lines: 3 proved, 1 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	✓ 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. The simplification of the integrand is algebraically correct, and the final definite integral evaluation is accurate.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the arc length formula. The simplification of the integrand is algebraically correct, and the final definite integral evaluation is accurate.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: fail (error) 2026-09-27 — The simplification of 1+(dy/dx)^2 is incorrect; the expression given is not a perfect square of the form shown, so the subsequent integral is based on a false integrand. The final result 2055/64 does not follow from the correct arc‑length integral.
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly sets up the arc length integral and computes the result. Although step 3 claims the expression simplifies to a specific perfect square form (which is algebraically incorrect for this function, as 1+(y')^2 is not a perfect square of a simple rational function), the final integration in step 4 yields the correct numerical answer, implying the CAS handled the integration correctly despite the misleading intermediate simplification claim. Since the final answer is correct and the setup is sound, it passes.
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.