The theorem of Pappus
Problem 5.585 · medium
The region under \( \displaystyle y = 3x \) for \( \displaystyle 0 \le x \le 1 \) 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
- \[ \int\limits_{0}^{1} 3 x\, dx = \frac{3}{2} \]The area.✓ Proved
- Its centroid is at ȳ = (1/A)∫ ½(mx)² dx = 1.Reviewed
- \[ \frac{2 \int\limits_{0}^{1} \frac{9 x^{2}}{2}\, dx}{3} = 1 \]ȳ = M_x / A.✓ Proved
- \[ 2 \pi 1 \cdot 3 \cdot \frac{1}{2} = 3 \pi \]V = 2π·d·A.✓ Proved
Answer \( 3 \pi \)
Lines: 3 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 |
| 5 | ✓ 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 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, and the centroid y-coordinate is correctly identified as 1 (the centroid of the triangle with vertices (0,0), (1,0), (1,3) is at y=1). The final volume calculation follows logically.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, and the centroid y-coordinate is correctly identified as 1 (the centroid of the triangle with vertices (0,0), (1,0), (1,3) is at y=1). The final volume calculation follows logically.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution incorrectly identifies the distance d as the y-coordinate of the centroid (ȳ = 1). For rotation about the x-axis, the distance from the centroid to the axis is indeed ȳ, but the calculation of ȳ is wrong. The centroid y-coordinate for the region under y=3x is ∫(1/2)(3x)^2 dx / A = (9/6)/(3/2) = 1. Wait, let me re-evaluate. ∫_0^1 (1/2)(3x)^2 dx = ∫_0^1 (9/2)x^2 dx = [3x^3]_0^1 = 3. Area A = 3/2. So ȳ = 3 / (3/2) = 2. The solution claims ȳ = 1. This is an error in the calculation of the centroid's y-coordinate. The integral of 9x^2/2 is 3x^3, evaluated from 0 to 1 is 3. 3 divided by 1.5 is 2. The solution says the integral result is 1, which is false.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, the centroid y-coordinate is correctly calculated as 1, and the final volume is derived accurately.gpt-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 2026-10-17 with SymPy 1.14.0.