Centers of mass and centroids
Problem 5.347 · medium
Find the centroid of the region bounded by \( \displaystyle y = x^{2} \) and \( \displaystyle y = 0 \) for \( \displaystyle 0 \le x \le 3 \).
- \[ \int\limits_{0}^{3} x^{2}\, dx = 9 \]The area.✓ Proved
- \[ \int\limits_{0}^{3} x^{3}\, dx = \frac{81}{4} \]The moment about the y-axis.✓ Proved
- \[ \int\limits_{0}^{3} \frac{x^{4}}{2}\, dx = \frac{243}{10} \]The moment about the x-axis: ∫ ½(top² − bottom²) dx.✓ Proved
- \[ \left[\begin{matrix}\frac{9}{4}\\\frac{27}{10}\end{matrix}\right] \]Divide each moment by the area.✓ Proved
Answer \( (\bar{x}, \bar{y}) = \left(\frac{9}{4}, \frac{27}{10}\right) \)
✓ 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 formulas for the area and moments of the region bounded by y=x^2 and y=0. The integrals are set up with the correct bounds and integrands, and the final division by the area yields the correct centroid coordinates.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the formulas for the area and moments of the region bounded by y=x^2 and y=0. The integrals are set up with the correct bounds and integrands, and the final division by the area yields the correct centroid coordinates.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the formulas for the area and moments of a region bounded by y=x^2 and y=0. The integrals are set up with the correct bounds and integrands, and the final division by area yields the correct centroid coordinates.
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 2026-10-07 with SymPy 1.14.0.