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The theorem of Pappus

Problem 5.656 · medium

The rectangle \( \displaystyle 2 \le x \le 5 \), \( \displaystyle 0 \le y \le 4 \) 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
\[ 3 \cdot 4 = 12 \]
Area = width × height.✓ Proved
The centroid is the center, at x = 7/2.
Reviewed
\[ 2 \pi 7 \cdot \frac{1}{2} \cdot 12 = 84 \pi \]
V = 2π·d·A.✓ Proved

Answer \( 84 \pi \)

Lines: 2 proved, 2 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	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 and centroid of the rectangle, and applies Pappus's Second Theorem accurately to find the volume.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer returned a non-object
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.