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Line integrals and work

Problem 12.7 · medium

Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the parabola y = x² from (0, 0) to (1, 1).

Parametrize: r(t) = ⟨t, t**2⟩ for 0 ≤ t ≤ 1.
Reviewed
\[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} t^{2}\end{matrix}\right] = \left[\begin{matrix}1\\2 t\end{matrix}\right] \]
r'(t).✓ Proved
\[ 2 t^{4} + t^{2} \]
F(r(t)) · r'(t).Reviewed
\[ \int\limits_{0}^{1} \left(2 t^{4} + t^{2}\right)\, dt = \frac{11}{15} \]
W = ∫ F · dr.✓ Proved

Answer \( \frac{11}{15} \)

✓ 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 parametrization is correct, the dot product calculation is accurate, and the integral evaluation is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The parametrization is correct, the dot product calculation is accurate, and the integral evaluation is correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The parametrization, dot product, and integration are all correct and clearly stated.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
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.