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The theorem of Pappus

Problem 5.768 · medium

The triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (2, 0) \) and \( \displaystyle (0, 3) \) 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
\[ 1 \cdot \frac{1}{2} \cdot 2 \cdot 3 = 3 \]
Area ½·base·height.✓ Proved
A triangle's centroid is the average of its vertices: ȳ = (0 + 0 + 3)/3 = 1.
Reviewed
\[ 2 \pi 1 \cdot 3 = 6 \pi \]
V = 2π·d·A.✓ Proved

Answer \( 6 \pi \)

✓ 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 applies the Theorem of Pappus. The area calculation is correct, and the centroid's y-coordinate (distance to the x-axis) is correctly identified as 1. The final volume calculation is accurate.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, and the centroid's y-coordinate (distance to the x-axis) is correctly identified as 1. The final volume calculation is accurate.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: pass 2026-11-03 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, and the centroid's distance from the axis of rotation (the y-coordinate of the centroid) is correctly identified as 1. The final volume calculation is accurate.
gpt-oss:20b: pass 2026-11-03
qwen3.6:27b-mlx: pass 2026-11-03 — The solution correctly identifies the area and the y-coordinate of the centroid, which serves as the distance to the axis of rotation. The application of Pappus's theorem is correct.
gpt-oss:20b: pass 2026-11-03

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 2026-11-04 with SymPy 1.14.0.