Mass from a density
Problem 5.234 · easy
A rod lies along \( \displaystyle 0 \le x \le 5 \) (meters) with linear density \( \displaystyle \rho(x) = 1 + e^{- x} \) kg/m. Find its mass.
- \[ \int\limits_{0}^{5} \left(1 + e^{- x}\right)\, dx = 6 - e^{-5} \]Mass = ∫ ρ(x) dx along the rod.✓ Proved
Answer \( 6 - e^{-5} \approx 5.9933 \)
✓ 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 solution correctly identifies mass as the integral of linear density over the rod's length and computes the definite integral accurately.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies mass as the integral of linear density over the rod's length and computes the definite integral accurately.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The setup correctly identifies mass as the integral of linear density over the length of the rod. The bounds and integrand are correct.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.