Line integrals and work
Problem 12.112 · medium
Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the line segment from (0, 0) to (3, 3).
- Parametrize: r(t) = ⟨3*t, 3*t⟩ for 0 ≤ t ≤ 1.Reviewed
- \[ \left[\begin{matrix}\frac{d}{d t} 3 t\\\frac{d}{d t} 3 t\end{matrix}\right] = \left[\begin{matrix}3\\3\end{matrix}\right] \]r'(t).✓ Proved
- \[ 27 t^{2} + 18 t = 9 t \left(3 t + 2\right) \]F(r(t)) · r'(t).✓ Proved
- \[ \int\limits_{0}^{1} 9 t \left(3 t + 2\right)\, dt = 18 \]W = ∫ F · dr.✓ Proved
Answer \( 18 \)
✓ 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, the dot product is computed correctly, and the integral evaluation is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The parametrization is correct, the dot product is computed correctly, and the integral evaluation is correct.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 (3,3). The dot product and integration are performed correctly, yielding the correct result of 18.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.