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The theorem of Pappus

Problem 5.581 · medium

The region under \( \displaystyle y = 1x \) for \( \displaystyle 0 \le x \le 1 \) is rotated about the \( \displaystyle x \)-axis. Use the theorem of Pappus to find the volume.

Pappus: V = A · 2πd, where d is the distance from the centroid to the axis.
Reviewed
\[ \int\limits_{0}^{1} x\, dx = \frac{1}{2} \]
The area.✓ Proved
Its centroid is at ȳ = (1/A)∫ ½(mx)² dx = 1/3.
Reviewed
\[ 2 \int\limits_{0}^{1} \frac{x^{2}}{2}\, dx = \frac{1}{3} \]
ȳ = M_x / A.✓ Proved
\[ 2 \pi 1 \cdot \frac{1}{3} \cdot 1 \cdot \frac{1}{2} = \frac{\pi}{3} \]
V = 2π·d·A.✓ Proved

Answer \( \frac{\pi}{3} \)

✓ Nihil obstat Lines: 3 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	a sentence; read, not computed
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ 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	the volume by disks or shells, numerically

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Theorem of Pappus, accurately calculates the area and the y-coordinate of the centroid, and derives the correct volume.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Theorem of Pappus, accurately calculates the area and the y-coordinate of the centroid, and derives the correct volume.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Theorem of Pappus, accurately calculates the area and the y-coordinate of the centroid, and derives the correct volume.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14
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/pappus, checked 2026-10-17 with SymPy 1.14.0.