Home›Calculus 2›The theorem of Pappus›Problem 5.590

The theorem of Pappus

Problem 5.590 · medium

The triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (5, 0) \) 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
\[ 1 \cdot \frac{1}{2} \cdot 5 \cdot 3 = \frac{15}{2} \]
Area ½·base·height.✓ Proved
A triangle's centroid is the average of its vertices: ȳ = (0 + 0 + 3)/3 = 1.
Reviewed
\[ 2 \pi 1 \cdot 15 \cdot \frac{1}{2} = 15 \pi \]
V = 2π·d·A.✓ Proved

Answer \( 15 \pi \)

✓ 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, accurately calculates the area and the y-coordinate of the centroid, and derives the correct volume.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Theorem of Pappus, accurately calculates the area and the y-coordinate of the centroid, and derives the correct volume.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, the centroid y-coordinate is correctly identified as the average of the y-vertices (which represents the distance to the x-axis), and the final volume calculation is accurate.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Theorem of Pappus. The area calculation is correct, the y-coordinate of the centroid is correctly identified as the distance to the x-axis, and the final volume calculation is accurate.
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.