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The theorem of Pappus

Problem 5.649 · medium

The rectangle \( \displaystyle 2 \le x \le 6 \), \( \displaystyle 0 \le y \le 1 \) is rotated about the \( \displaystyle y \)-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
\[ 4 \cdot 1 = 4 \]
Area = width × height.✓ Proved
The centroid is the center, at x = 4.
Reviewed
\[ 2 \pi 4 \cdot 4 = 32 \pi \]
V = 2π·d·A.✓ Proved

Answer \( 32 \pi \)

✓ 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	a sentence; read, not computed
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	the volume by disks or shells, numerically

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the area, the centroid's distance from the axis of rotation, and applies Pappus's Second Theorem accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the area, the centroid's distance from the axis of rotation, and applies Pappus's Second Theorem accurately.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the area, the centroid's distance from the axis of rotation, and applies Pappus's Second Theorem accurately.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the area and centroid of the rectangle, applies Pappus's Second Theorem with the correct distance from the centroid to the axis of rotation, and calculates the volume accurately.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.