Centers of mass and centroids
Problem 5.273 · medium
Find the centroid of the region bounded by \( \displaystyle y = 4 - x \) and \( \displaystyle y = 0 \) for \( \displaystyle 0 \le x \le 2 \).
- \[ \int\limits_{0}^{2} \left(4 - x\right)\, dx = 6 \]The area.✓ Proved
- \[ \int\limits_{0}^{2} x \left(4 - x\right)\, dx = \frac{16}{3} \]The moment about the y-axis.✓ Proved
- \[ \int\limits_{0}^{2} \frac{\left(4 - x\right)^{2}}{2}\, dx = \frac{28}{3} \]The moment about the x-axis: ∫ ½(top² − bottom²) dx.✓ Proved
- \[ \left[\begin{matrix}\frac{8}{9}\\\frac{14}{9}\end{matrix}\right] \]Divide each moment by the area.✓ Proved
Answer \( (\bar{x}, \bar{y}) = \left(\frac{8}{9}, \frac{14}{9}\right) \)
Every line of this solution was proved by the computer algebra system SymPy. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | numerical quadrature of each moment |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the formulas for the area and moments, sets up the integrals with the correct bounds and functions, and performs the division to find the centroid coordinates. The algebraic results are verified as correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the formulas for the area and moments, sets up the integrals with the correct bounds and functions, and performs the division to find the centroid coordinates. The algebraic results are verified as correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The calculation for the moment about the x-axis is incorrect. The integral of (4-x)^2/2 from 0 to 2 is 28/3, but dividing this by the area (6) yields 14/9, which is correct for y-bar. Wait, let me re-calculate. Integral of (4-x)^2/2 dx from 0 to 2. Let u = 4-x, du = -dx. Limits 4 to 2. Integral u^2/2 du = u^3/6. Evaluated from 2 to 4: (64/6 - 8/6) = 56/6 = 28/3. Area is 6. y_bar = (28/3) / 6 = 28/18 = 14/9. The calculation is actually correct. Let me re-read the prompt. 'Find the centroid...'. The solution steps are: 1. Area = 6. Correct. 2. M_y = 16/3. x_bar = (16/3)/6 = 16/18 = 8/9. Correct. 3. M_x = 28/3. y_bar = (28/3)/6 = 14/9. Correct. 4. Result matches. Why would it fail? Ah, I need to check if the setup models the problem correctly. The region is bounded by y=4-x, y=0, 0<=x<=2. This is a trapezoid with vertices (0,0), (2,0), (2,2), (0,4). Area = (4+2)*2/2 = 6. Correct. Centroid of trapezoid: x_bar = (2/3) * (a^2 + ab + b^2) / (a+b) * h? No, simpler to use integrals. The integrals are set up correctly. The arithmetic is correct. The final answer is correct. Is there a subtle error? 'Matrix([...]) = Matrix([...])' is a weird way to state the final answer, but the math is sound. Wait, looking at step 3 comment: 'The moment about the x-axis: ∫ ½(top² − bottom²) dx'. This is the correct formula for M_x. The integral evaluation is correct. The division is correct. The result is correct. Why did I think it might fail? Let me double check the integral of x(4-x). 4x - x^2. Integral 2x^2 - x^3/3. From 0 to 2: 2(4) - 8/3 = 8 - 8/3 = 16/3. Correct. Integral of (4-x)^2/2. (1/2)(16 - 8x + x^2). Integral (1/2)(16x - 4x^2 + x^3/3). From 0 to 2: (1/2)(32 - 16 + 8/3) = (1/2)(16 + 8/3) = 8 + 4/3 = 28/3. Correct. Everything seems correct. Is there a 'style' issue? The prompt asks to find what is wrong. If nothing is wrong, I should say pass. But usually these adversarial reviews have a trick. Let's look at the bounds. 0 <= x <= 2. y=4-x. At x=0, y=4. At x=2, y=2. The region is above y=0. Correct. Maybe the 'Matrix' line is considered 'style' or 'misleading' because it doesn't explicitly state the centroid coordinates but rather an equality of matrices? Or is it just a weird format? The prompt says 'Stated answer: (x_bar, y_bar) = ...'. The solution ends with a matrix equality. This is a bit odd but not mathematically wrong. However, looking closely at the provided solution text: '4. [equation, unchecked] Matrix([...]) = Matrix([...]) -- Divide each moment by the area.' This is a valid check. I will mark it as pass unless I find a hidden error. Wait, is the centroid of a trapezoid with parallel vertical sides? No, parallel horizontal sides? No, the parallel sides are vertical? No, the region is bounded by x=0, x=2, y=0, y=4-x. The sides x=0 and x=2 are vertical. The top is slanted. The bottom is horizontal. It is a trapezoid with height 2 (along x-axis) and parallel vertical sides of length 4 and 2? No, the parallel sides are the vertical lines x=0 and x=2? No, the definition of a trapezoid is one pair of parallel sides. Here y=0 is horizontal. y=4-x is not horizontal. x=0 and x=2 are vertical. So the sides x=0 and x=2 are parallel. So it is a trapezoid with height 2 (distance between parallel sides) and bases 4 and 2. The centroid of such a trapezoid (with parallel sides vertical) is not the standard formula. But the integral method is robust. The integrals are correct. The result is correct. I will pass it.gpt-oss:20b: pass 2026-10-05
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/center_of_mass, checked 2026-10-05 with SymPy 1.14.0.