Centers of mass and centroids
Problem 5.205 · medium
Find the centroid of the region bounded by \( \displaystyle y = 2 x \) and \( \displaystyle y = x^{2} \).
- \[ \int\limits_{0}^{2} \left(- x^{2} + 2 x\right)\, dx = \frac{4}{3} \]The area.✓ Proved
- \[ \int\limits_{0}^{2} x \left(- x^{2} + 2 x\right)\, dx = \frac{4}{3} \]The moment about the y-axis.✓ Proved
- \[ \int\limits_{0}^{2} \left(- \frac{x^{4}}{2} + 2 x^{2}\right)\, dx = \frac{32}{15} \]The moment about the x-axis: ∫ ½(top² − bottom²) dx.✓ Proved
- \[ \left[\begin{matrix}1\\\frac{8}{5}\end{matrix}\right] \]Divide each moment by the area.✓ Proved
Answer \( (\bar{x}, \bar{y}) = \left(1, \frac{8}{5}\right) \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 bounds, area, and moments, and correctly applies the centroid formulas. The comments accurately describe the integrals used.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the bounds, area, and moments, and correctly applies the centroid formulas. The comments accurately describe the integrals used.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the bounds, area, and moments, and the final centroid coordinates are correct.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.