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The theorem of Pappus

Problem 5.644 · medium

The disk \( \displaystyle (x - 5)^2 + y^2 \le 4 \) is rotated about the \( \displaystyle y \)-axis. Use the theorem of Pappus to find the volume of the torus.

Pappus: V = A · 2πd, where d is the distance from the centroid to the axis.
Reviewed
\[ 2^{2} \pi = 4 \pi \]
The disk's area, πa².✓ Proved
The centroid is the center (5, 0), 5 from the axis.
Reviewed
\[ 2 \pi 5 \cdot 4 \pi = 40 \pi^{2} \]
V = 2π·d·A.✓ Proved

Answer \( 40 \pi^{2} \)

✓ 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, identifying the area of the disk and the distance of its centroid from the axis of rotation. The calculation is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the Theorem of Pappus, identifying the area of the disk and the distance of its centroid from the axis of rotation. The calculation is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Theorem of Pappus, identifying the area of the disk and the distance of its centroid from the axis of rotation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the area of the disk, the location of the centroid, and the distance to the axis of rotation. The application of Pappus's Second Theorem is correct and yields the right result.
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.