Line integrals and work
Problem 12.107 · medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the line segment from (0, 0) to (3, 1).
- Parametrize: r(t) = ⟨3*t, t⟩ for 0 ≤ t ≤ 1.Reviewed
- \[ \left[\begin{matrix}\frac{d}{d t} 3 t\\\frac{d}{d t} t\end{matrix}\right] = \left[\begin{matrix}3\\1\end{matrix}\right] \]r'(t).✓ Proved
- \[ 30 t^{2} \]F(r(t)) · r'(t).✓ Proved
- \[ \int\limits_{0}^{1} 30 t^{2}\, dt = 10 \]W = ∫ F · dr.✓ Proved
Answer \( 10 \)
✓ 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 (3,1). The dot product and integration are correctly calculated, leading to the correct answer of 10.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The parametrization is correct for the line segment from (0,0) to (3,1). The dot product and integration are correctly calculated, leading to the correct answer of 10.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The parametrization is correct for the line segment from (0,0) to (3,1). The calculation of the dot product and the subsequent integration are correct, yielding the stated answer of 10.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.