Line integrals and work
Problem 12.45 · medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the line segment from (0, 0) to (1, 3).
- Parametrize: r(t) = ⟨t, 3*t⟩ for 0 ≤ t ≤ 1.Reviewed
- \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} 3 t\end{matrix}\right] = \left[\begin{matrix}1\\3\end{matrix}\right] \]r'(t).✓ Proved
- \[ - 2 t \]F(r(t)) · r'(t).Reviewed
- \[ \int\limits_{0}^{1} \left(- 2 t\right)\, dt = -1 \]W = ∫ F · dr.✓ Proved
Answer \( -1 \)
Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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 (1,3). The computation of F(r(t)) and the dot product with r'(t) are correct, leading to the correct integral and final answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The parametrization is correct for the line segment from (0,0) to (1,3). The computation of F(r(t)) and the dot product with r'(t) are correct, leading to the correct integral and final answer.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The parametrization is correct for the line segment from (0,0) to (1,3). The computation of the dot product and the integral are correct, leading to the stated answer of -1.gpt-oss:20b: fail (misleading) 2026-09-27 — The solution claims that the dot product F(r(t))·r'(t) equals "-2*t, -2*t", which incorrectly presents a vector instead of the scalar -2t. While the subsequent integral is correct, the sentence misleads a student about the nature of the dot product.qwen3.6:27b-mlx: pass 2026-09-26 — The parametrization is correct, the dot product calculation is correct, 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 2026-10-13 with SymPy 1.14.0.