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Line integrals and work

Problem 12.111 · medium

Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) 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
\[ 0 \]
F(r(t)) · r'(t).Reviewed
\[ \int\limits_{0}^{1} 0\, dt = 0 \]
W = ∫ F · dr.✓ Proved

Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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: pass
qwen3.6:27b-mlx: pass — The solution correctly parametrizes the line segment, computes the dot product as zero, and integrates to find the work is zero. The steps are logically sound and mathematically correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly parametrizes the line segment, computes the dot product as zero, and integrates to find the work is zero. The steps are logically sound and mathematically correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The parametrization is correct for the line segment from (0,0) to (3,3). The calculation of the dot product F(r(t)) · r'(t) yielding 0 is correct because the vector field is orthogonal to the path. The integral of 0 is 0.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The parametrization is correct, the dot product calculation F(r(t)) · r'(t) = (-3t)(3) + (3t)(3) = 0 is correct, and the integral of 0 is 0. The solution is sound.
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 2026-10-13 with SymPy 1.14.0.