Centers of mass and centroids
Problem 5.422 · medium
A rod on \( \displaystyle 0 \le x \le 2 \) has density \( \displaystyle \rho(x) = 2 x + 2 \). Find its center of mass.
- \[ \int\limits_{0}^{2} \left(2 x + 2\right)\, dx = 8 \]The mass.✓ Proved
- \[ \int\limits_{0}^{2} x \left(2 x + 2\right)\, dx = \frac{28}{3} \]The moment about x = 0.✓ Proved
- \[ \frac{7}{6} \]x̄ = moment / mass.✓ Proved
Answer \( \bar{x} = \frac{7}{6} \)
✓ 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the definition of center of mass for a 1D rod, with correct bounds, density function, and arithmetic.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the definition of center of mass for a 1D rod, with correct bounds, density function, and arithmetic.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the formulas for mass and moment, computes the integrals accurately, and derives the correct center of mass.
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 2026-10-09 with SymPy 1.14.0.