Centers of mass and centroids
Problem 5.209 · medium
Find the centroid of the region bounded by \( \displaystyle y = 4 - x^{2} \) and \( \displaystyle y = 0 \) for \( \displaystyle -2 \le x \le 2 \).
- \[ \int\limits_{-2}^{2} \left(4 - x^{2}\right)\, dx = \frac{32}{3} \]The area.✓ Proved
- \[ \int\limits_{-2}^{2} x \left(4 - x^{2}\right)\, dx = 0 \]The moment about the y-axis.✓ Proved
- \[ \int\limits_{-2}^{2} \frac{\left(4 - x^{2}\right)^{2}}{2}\, dx = \frac{256}{15} \]The moment about the x-axis: ∫ ½(top² − bottom²) dx.✓ Proved
- \[ \left[\begin{matrix}0\\\frac{8}{5}\end{matrix}\right] \]Divide each moment by the area.✓ Proved
Answer \( (\bar{x}, \bar{y}) = \left(0, \frac{8}{5}\right) \)
✓ 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 area and moments, applies the centroid formulas accurately, and arrives at the correct result. The steps are logically sound and mathematically correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the area and moments, applies the centroid formulas accurately, and arrives at the correct result. The steps are logically sound and mathematically correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly sets up and computes the area and moments for the centroid of the region bounded by y = 4 - x^2 and y = 0. The formulas used are standard and applied correctly.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.