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Arc length of x = g(y)

Problem 5.562 · hard

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

For x = g(y), L = ∫ √(1 + (dx/dy)²) dy.
\[ \frac{d}{d y} \left(\frac{y^{3}}{6} + \frac{1}{2 y}\right) = \frac{y^{2}}{2} - \frac{1}{2 y^{2}} \]
dx/dy.✓ Proved
\[ \left(\frac{d}{d y} \left(\frac{y^{3}}{6} + \frac{1}{2 y}\right)\right)^{2} + 1 = \frac{\left(y^{4} + 1\right)^{2}}{4 y^{4}} \]
1 + (dx/dy)², simplified.✓ Proved
\[ \int\limits_{1}^{2} \frac{\frac{y^{4}}{2} + \frac{1}{2}}{y^{2}}\, dy = \frac{17}{12} \]
Integrate.✓ Proved

Answer \( \frac{17}{12} \)

Lines: 3 proved, 1 not checked. 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	Not checked	—	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 20,000-segment polyline along the curve

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly take the square root of the squared derivative term before integrating. Step 3 computes 1 + (dx/dy)^2, but Step 4 integrates (y^4 + 1)/(2y^2), which is the square root of the expression in Step 3, without stating this crucial algebraic step. This omission hides the non-trivial simplification that makes the integral solvable and could mislead a student into thinking the square root disappears or is irrelevant.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to explicitly take the square root of the squared derivative term before integrating. Step 3 computes 1 + (dx/dy)^2, but Step 4 integrates (y^4 + 1)/(2y^2), which is the square root of the expression in Step 3, without stating this crucial algebraic step. This omission hides the non-trivial simplification that makes the integral solvable and could mislead a student into thinking the square root disappears or is irrelevant.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to take the square root of the squared derivative term. Step 3 computes 1 + (dx/dy)^2, but Step 4 integrates the unsimplified numerator of that fraction (y^4/2 + 1/2) divided by y^2, rather than integrating the square root, which is (y^2/2 + 1/(2y^2)). Although the final numerical answer happens to be correct due to a coincidence in the specific bounds and function, the method shown is mathematically invalid for arc length.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly take the square root of 1 + (dx/dy)^2 before integrating. While the integrand in step 4 happens to be the correct result of the square root, the logical step of simplifying sqrt((y^4 + 1)^2 / (4y^4)) to (y^2 + 1/y^2)/2 is missing, making the derivation incomplete and potentially confusing regarding the handling of the absolute value.
gpt-oss:20b: pass 2026-10-14

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_y, checked 2026-10-17 with SymPy 1.14.0.