Home›Calculus 2›The theorem of Pappus›Problem 5.618

The theorem of Pappus

Problem 5.618 · medium

The region under \( \displaystyle y = 4x \) 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
\[ \int\limits_{0}^{1} 4 x\, dx = 2 \]
The area.✓ Proved
Its centroid is at ȳ = (1/A)∫ ½(mx)² dx = 4/3.
Reviewed
\[ \frac{\int\limits_{0}^{1} 8 x^{2}\, dx}{2} = \frac{4}{3} \]
ȳ = M_x / A.✓ Proved
\[ 2 \pi 4 \cdot \frac{1}{3} \cdot 2 = \frac{16 \pi}{3} \]
V = 2π·d·A.✓ Proved

Answer \( \frac{16 \pi}{3} \)

✓ Nihil obstat Lines: 3 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
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: pass — The solution correctly applies the Second Theorem of Pappus. The area and centroid calculations are correct, and the final volume is derived accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the Second Theorem of Pappus. The area and centroid calculations are correct, and the final volume is derived accurately.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the Second Theorem of Pappus. The area calculation, centroid y-coordinate derivation, and final volume computation are all accurate and logically sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution incorrectly identifies the distance d as the y-coordinate of the centroid (4/3). Since the region is rotated about the x-axis, the distance from the centroid to the axis of rotation is indeed ȳ, but the formula application in step 5 uses d=4/3 and A=2, yielding 16π/3. Wait, let's re-read carefully. Step 1 says V = A * 2πd. Step 5 calculates 2*pi*(4/3)*2. This is correct algebraically. However, Step 3 states the formula for ȳ as (1/A)∫ ½(mx)² dx. This is notationally confusing ('mx' is undefined, likely meant y(x)). More importantly, the problem asks to use Pappus. The setup is correct. Let's look closer at Step 3. It says 'Its centroid is at ȳ = ...'. The calculation in Step 4 confirms ȳ = 4/3. The volume is A * 2π * ȳ = 2 * 2π * (4/3) = 16π/3. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's constraints. 'Find what is wrong'. Is there a subtle error? Step 3: 'ȳ = (1/A)∫ ½(mx)² dx'. 'mx' is not defined. It should be y(x) or f(x). This is a notation error. Is it 'error' or 'style'? It's a notation error that makes the formula look like it depends on an undefined variable 'm'. A student might think 'm' is a parameter. It's misleading/wrong notation. Let's look at Step 1. 'd is the distance from the centroid to the axis'. Correct. Step 3. The formula for the moment M_x is ∫ (1/2) [f(x)]^2 dx. The text writes '½(mx)²'. If 'mx' is a typo for 'y(x)' or 'f(x)', it's a style issue. But if it implies a linear function mx, it's ambiguous. However, looking at Step 4, it integrates 8*x**2. 8*x**2 comes from 2 * (4x)^2? No. M_x = ∫_0^1 (1/2)(4x)^2 dx = ∫_0^1 8x^2 dx. The integral of 8x^2 is 8/3. Divided by A=2 gives 4/3. The math is correct. The notation 'mx' in step 3 is the only flaw. Is it severe enough to fail? 'mx' is not standard. It looks like a typo for 'y' or 'f(x)'. I will mark it as error because 'mx' is undefined and misleading.
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.