Line integrals and work
Problem 12.84 · medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (1, 2).
- Parametrize: r(t) = ⟨t, 2*t⟩ for 0 ≤ t ≤ 1.Reviewed
- \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} 2 t\end{matrix}\right] = \left[\begin{matrix}1\\2\end{matrix}\right] \]r'(t).✓ Proved
- \[ t \]F(r(t)) · r'(t).✓ Proved
- \[ \int\limits_{0}^{1} t\, dt = \frac{1}{2} \]W = ∫ F · dr.✓ Proved
Answer \( \frac{1}{2} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ 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 | a 20,000-chord polygon along the path gives the same work |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The parametrization is correct for the line segment from (0,0) to (1,2). The calculation of the dot product F(r(t)) · r'(t) and the subsequent integration are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The parametrization is correct for the line segment from (0,0) to (1,2). The calculation of the dot product F(r(t)) · r'(t) and the subsequent integration are correct.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The parametrization is correct, the dot product calculation is accurate, and the integration yields the correct result.gpt-oss:20b: pass 2026-09-26
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/line_integral_work, checked 2026-09-26 with SymPy 1.14.0.