Centers of mass and centroids
Problem 5.280 · medium
A rod on \( \displaystyle 0 \le x \le 1 \) has density \( \displaystyle \rho(x) = x^{2} + 4 \). Find its center of mass.
- \[ \int\limits_{0}^{1} \left(x^{2} + 4\right)\, dx = \frac{13}{3} \]The mass.✓ Proved
- \[ \int\limits_{0}^{1} x \left(x^{2} + 4\right)\, dx = \frac{9}{4} \]The moment about x = 0.✓ Proved
- \[ \frac{27}{52} \]x̄ = moment / mass.✓ Proved
Answer \( \bar{x} = \frac{27}{52} \)
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: fail (error) — The final answer is incorrect. The center of mass is the moment divided by the mass, which is (9/4) / (13/3) = 27/52. Wait, 9/4 * 3/13 = 27/52. The calculation is actually correct. Let me re-read the prompt's stated answer. Stated answer: 27/52. My calculation: 27/52. The solution is correct. Why did I think it was wrong? Ah, I need to check the setup. Density rho(x) = x^2 + 4. Mass M = integral_0^1 (x^2+4) dx = [x^3/3 + 4x]_0^1 = 1/3 + 4 = 13/3. Moment M_x = integral_0^1 x(x^2+4) dx = integral_0^1 (x^3+4x) dx = [x^4/4 + 2x^2]_0^1 = 1/4 + 2 = 9/4. x_bar = (9/4) / (13/3) = 27/52. The solution is correct. The verdict should be pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The final answer is incorrect. The center of mass is the moment divided by the mass, which is (9/4) / (13/3) = 27/52. Wait, 9/4 * 3/13 = 27/52. The calculation is actually correct. Let me re-read the prompt's stated answer. Stated answer: 27/52. My calculation: 27/52. The solution is correct. Why did I think it was wrong? Ah, I need to check the setup. Density rho(x) = x^2 + 4. Mass M = integral_0^1 (x^2+4) dx = [x^3/3 + 4x]_0^1 = 1/3 + 4 = 13/3. Moment M_x = integral_0^1 x(x^2+4) dx = integral_0^1 (x^3+4x) dx = [x^4/4 + 2x^2]_0^1 = 1/4 + 2 = 9/4. x_bar = (9/4) / (13/3) = 27/52. The solution is correct. The verdict should be pass.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The calculated center of mass is 27/13, not 27/52. The solution incorrectly computes the ratio of the moment (9/4) to the mass (13/3) as 27/52 instead of 27/13.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.