Home›Calculus 2›The theorem of Pappus›Problem 5.619

The theorem of Pappus

Problem 5.619 · medium

The region under $ \displaystyle y = 2x $ 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.
\[ \int\limits_{0}^{1} 2 x\, dx = 1 \]
The area.✓ Proved
Its centroid is at ȳ = (1/A)∫ ½(mx)² dx = 2/3.
\[ \int\limits_{0}^{1} 2 x^{2}\, dx = \frac{2}{3} \]
ȳ = M_x / A.✓ Proved
\[ 2 \pi 2 \cdot \frac{1}{3} \cdot 1 = \frac{4 \pi}{3} \]
V = 2π·d·A.✓ Proved

Answer $ \frac{4 \pi}{3} $

Lines: 3 proved, 2 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the distance $d$ in Pappus's theorem as the y-coordinate of the centroid ($\\bar{y}$). Since the region is rotat

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the distance $d$ in Pappus's theorem as the y-coordinate of the centroid ($\\bar{y}$). Since the region is rotat
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Theorem of Pappus, accurately calculating the area and the y-coordinate of the centroid for the region under y=2x. The final volume calculation is correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The formula for the y-coordinate of the centroid is incorrect; it should be (1/A)∫ (1/2)f(x)^2 dx, but the solution writes ∫ ½(mx)^2 dx where m is undefined and implies a linear relationship that doesn't match the function y=2x directly in the notation used. More critically, the calculation in line 4 uses 2*x**2 which corresponds to (2x)^2, but the coefficient 1/2 is missing from the integral expression in line 3's text description, although the numerical result 2/3 is correct for the centroid of a triangle with height 2. However, the text in line 3 says '∫ ½(mx)² dx' which is ambiguous and potentially misleading if m is not defined as the slope. Let's look closer. Line 3 says '∫ ½(mx)² dx'. If y=mx, then y=2x, so m=2. The integral of 1/2 * (2x)^2 is integral of 2x^2. The result is 2/3. The text is slightly sloppy with 'm' but the math holds. Wait, is there a bigger error? Pappus theorem: V = 2 * pi * d * A. d is the distance from centroid to axis. For rotation about x-axis, d = y_bar. y_bar = 2/3. A = 1. V = 2 * pi * (2/3) * 1 = 4pi/3. The solution is actually correct. Why would it fail? Let's re-read carefully. 'Its centroid is at ȳ = (1/A)∫ ½(mx)² dx = 2/3.' The variable 'm' is not defined. In calculus contexts, this is a style issue or a minor error if m is assumed to be the slope. But is it 'error' or 'style'? The prompt asks to find what is wrong. The notation 'm' is undefined. However, the calculation is correct. Is there a subtle error? The region is under y=2x. Centroid y-coordinate is indeed 2/3. Volume is 4pi/3. The solution is correct. I will mark it pass.
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 2026-10-18 with SymPy 1.14.0.