Centers of mass and centroids
Problem 5.348 · medium
A rod on \( \displaystyle 0 \le x \le 5 \) has density \( \displaystyle \rho(x) = x^{2} + 2 \). Find its center of mass.
- \[ \int\limits_{0}^{5} \left(x^{2} + 2\right)\, dx = \frac{155}{3} \]The mass.✓ Proved
- \[ \int\limits_{0}^{5} x \left(x^{2} + 2\right)\, dx = \frac{725}{4} \]The moment about x = 0.✓ Proved
- \[ \frac{435}{124} \]x̄ = moment / mass.✓ Proved
Answer \( \bar{x} = \frac{435}{124} \)
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)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the formulas for mass and moment, computes the integrals accurately, and derives the correct center of mass.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The stated answer 435/124 is incorrect. The correct center of mass is (725/4) / (155/3) = 2175/620 = 435/124? Wait, 725/4 * 3/155 = (725*3)/(4*155). 725/155 = 145/31. So 145*3 / (4*31) = 435/124. The arithmetic is actually correct. Let me re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. The solution provided contains NO sentences. It only contains equations. The prompt asks to review 'worked calculus solutions' and specifically says 'The sentences are what you are here for.' However, if there are no sentences, can I fail it? The prompt says 'Check that... the setup models the problem... each theorem... each conclusion follows'. Line 3 is an equation '435/124 = 435/124' with a comment '-- x̄ = moment / mass.' This is not a sentence explaining the step, it's a tautology with a comment. The solution lacks the necessary explanatory sentences to justify the steps (e.g., defining mass and moment integrals). But more importantly, is the math right? Mass = int_0^5 (x^2+2) dx = [x^3/3 + 2x]_0^5 = 125/3 + 10 = 155/3. Correct. Moment = int_0^5 x(x^2+2) dx = int_0^5 (x^3+2x) dx = [x^4/4 + x^2]_0^5 = 625/4 + 25 = 625/4 + 100/4 = 725/4. Correct. x_bar = (725/4) / (155/3) = 725/4 * 3/155. 725/155 = 145/31. 145*3 / (4*31) = 435/124. The math is correct. The format is just equations. The prompt says 'Reply with JSON... Say pass with empty notes if it is sound.' Is it sound? It's a valid derivation, just lacking text. But wait, line 3 is '435/124 = 435/124'. This is a tautology, not a derivation step. A proper step would be 'x_bar = (725/4) / (155/3) = 435/124'. The line provided doesn't show the division. It just asserts the final answer equals itself. This is a 'style' or 'misleading' issue because it doesn't show the work of dividing the moment by the mass. However, usually 'pass' is for sound solutions. Is it sound? The logic is missing. I will mark it as fail due to missing logical connection in the final step (the equation doesn't represent the calculation performed).gpt-oss:20b: pass 2026-10-07
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.