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Line integrals and work

Problem 12.37 · medium

Find the work done by \( \displaystyle \mathbf{F} = \langle y^{2}, 2 x \rangle \) along the unit circle, counterclockwise.

Parametrize: r(t) = ⟨cos(t), sin(t)⟩ for 0 ≤ t ≤ 2*pi.
Reviewed
\[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(t \right)}\\\cos{\left(t \right)}\end{matrix}\right] \]
r'(t).✓ Proved
\[ - \sin^{3}{\left(t \right)} + 2 \cos^{2}{\left(t \right)} \]
F(r(t)) · r'(t).Reviewed
\[ \int\limits_{0}^{2 \pi} \left(- \sin^{3}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}\right)\, dt = 2 \pi \]
W = ∫ F · dr.✓ Proved

Answer \( 2 \pi \)

✓ 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, dot product, and integration are all correct. The solution is sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The parametrization, dot product, and integration are all correct. The solution is sound.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The parametrization, dot product calculation, and integration are all correct. The solution is sound.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The parametrization is correct for the unit circle counterclockwise. The dot product and integration steps are algebraically verified and yield 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.