Mass from a density
Problem 5.382 · easy
A rod lies along \( \displaystyle 0 \le x \le 4 \) (meters) with linear density \( \displaystyle \rho(x) = 4 x + 1 \) kg/m. Find its mass.
- \[ \int\limits_{0}^{4} \left(4 x + 1\right)\, dx = 36 \]Mass = ∫ ρ(x) dx along the rod.✓ Proved
Answer \( 36 \approx 36 \)
✓ 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 |
| 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 setup correctly identifies mass as the integral of linear density over the length of the rod. The bounds and integrand are correct, and the calculation is verified.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The setup correctly identifies mass as the integral of linear density over the length of the rod. The bounds and integrand are correct, and the calculation is verified.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The setup correctly identifies mass as the integral of linear density over the length of the rod. The bounds and integrand are correct.
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/mass_density, checked 2026-10-08 with SymPy 1.14.0.