Centers of mass and centroids
Problem 5.488 · medium
A rod on \( \displaystyle 0 \le x \le 2 \) has density \( \displaystyle \rho(x) = x^{2} + 3 \). Find its center of mass.
- \[ \int\limits_{0}^{2} \left(x^{2} + 3\right)\, dx = \frac{26}{3} \]The mass.✓ Proved
- \[ \int\limits_{0}^{2} x \left(x^{2} + 3\right)\, dx = 10 \]The moment about x = 0.✓ Proved
- \[ \frac{15}{13} \]x̄ = moment / mass.✓ Proved
Answer \( \bar{x} = \frac{15}{13} \)
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 |
| 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 identifies the formulas for mass and moment, computes the integrals accurately, and derives the correct center of mass.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the formulas for mass and moment, computes the integrals accurately, and derives the correct center of mass.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The calculated center of mass is 15/13, but the stated answer is 15/13. Wait, 10 / (26/3) = 30/26 = 15/13. The calculation is correct. Why did I think it was wrong? Let me re-read the prompt. 'Stated answer: \bar{x} = \frac{15}{13}'. The solution derives 15/13. The steps are correct. The setup is correct. The algebra is checked by the CAS. The sentences are minimal but correct. Is there a hidden trap? 'A rod on 0 <= x <= 2'. Density x^2 + 3. Mass = int_0^2 (x^2+3) dx = [x^3/3 + 3x]_0^2 = 8/3 + 6 = 26/3. Moment = int_0^2 x(x^2+3) dx = int_0^2 (x^3+3x) dx = [x^4/4 + 3x^2/2]_0^2 = 16/4 + 12/2 = 4 + 6 = 10. x_bar = 10 / (26/3) = 30/26 = 15/13. Everything is correct. The verdict should be pass.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.