The theorem of Pappus
Problem 5.624 · medium
The disk \( \displaystyle (x - 7)^2 + y^2 \le 9 \) 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
- \[ 3^{2} \pi = 9 \pi \]The disk's area, πa².✓ Proved
- The centroid is the center (7, 0), 7 from the axis.Reviewed
- \[ 2 \pi 7 \cdot 9 \pi = 126 \pi^{2} \]V = 2π·d·A.✓ Proved
Answer \( 126 \pi^{2} \)
✓ 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: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-14gpt-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 2026-10-18 with SymPy 1.14.0.