Line integrals and work
Problem 12.1 · medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x^{2}, x y \rangle \) along the unit circle, counterclockwise.
- Parametrize: r(t) = ⟨cos(t), sin(t)⟩ for 0 ≤ t ≤ 2*pi.
- \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(t \right)}\\\cos{\left(t \right)}\end{matrix}\right] \]r'(t).✓ Proved
- \[ 0 \]F(r(t)) · r'(t).
- \[ \int\limits_{0}^{2 \pi} 0\, dt = 0 \]W = ∫ F · dr.✓ Proved
Answer \( 0 \)
Lines: 2 proved, 2 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | 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: fail (error) — The dot product F(r(t)) · r'(t) is calculated as 0, but it is actually -x^2 sin(t) + x y cos(t) = -cos^3(t) + cos^2(t)sin^2(t), which is not identically zero. The work is not zero.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The dot product F(r(t)) · r'(t) is calculated as 0, but it is actually -x^2 sin(t) + x y cos(t) = -cos^3(t) + cos^2(t)sin^2(t), which is not identically zero. The work is not zero.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The dot product calculation in step 3 is incorrect. F(r(t)) is <cos^2(t), cos(t)sin(t)>, and r'(t) is <-sin(t), cos(t)>. The dot product is -cos^2(t)sin(t) + cos^2(t)sin(t) = 0. Wait, let me re-evaluate. F = <x^2, xy>. r(t) = <cos t, sin t>. F(r(t)) = <cos^2 t, cos t sin t>. r'(t) = <-sin t, cos t>. Dot product: (cos^2 t)(-sin t) + (cos t sin t)(cos t) = -cos^2 t sin t + cos^2 t sin t = 0. The calculation is actually correct. The integral of 0 is 0. The answer is correct. Why did I think it was wrong? Let me double check Green's Theorem. curl F = d/dx(xy) - d/dy(x^2) = y - 0 = y. Integral of y over unit disk. By symmetry, integral of y over unit disk is 0. So the answer 0 is correct. The steps are correct. Step 3 says '0, 0 -- F(r(t)) . r'(t)'. This notation is slightly ambiguous but likely means the result is 0. Step 4 integrates 0 to get 0. The solution is sound.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The dot product calculation in step 3 is incorrect. F(r(t)) = <cos^2(t), cos(t)sin(t)> and r'(t) = <-sin(t), cos(t)>, so F · r' = -cos^2(t)sin(t) + cos^2(t)sin(t) = 0 is actually correct, wait. Let me re-evaluate. F = <x^2, xy>. r(t) = <cos t, sin t>. F(r(t)) = <cos^2 t, cos t sin t>. r'(t) = <-sin t, cos t>. Dot product: (cos^2 t)(-sin t) + (cos t sin t)(cos t) = -cos^2 t sin t + cos^2 t sin t = 0. The calculation is actually correct. The integral of 0 is 0. The answer is 0. Why did I think it was wrong? Let me check Green's Theorem. curl F = d/dx(xy) - d/dy(x^2) = y - 0 = y. Integral of y over unit disk. By symmetry, integral of y over symmetric region is 0. So the answer 0 is correct. The steps are correct. Step 3 says '0, 0 -- F(r(t)) · r'(t)'. This notation is slightly ambiguous but the result 0 is correct. Is there a subtle error? No. The solution is sound.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.