Line integrals and work
Problem 12.5 · medium
Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the parabola y = x² from (0, 0) to (1, 1).
- Parametrize: r(t) = ⟨t, t**2⟩ for 0 ≤ t ≤ 1.
- \[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} t^{2}\end{matrix}\right] = \left[\begin{matrix}1\\2 t\end{matrix}\right] \]r'(t).✓ Proved
- \[ t^{2} \]F(r(t)) · r'(t).
- \[ \int\limits_{0}^{1} t^{2}\, dt = \frac{1}{3} \]W = ∫ F · dr.✓ Proved
Answer \( \frac{1}{3} \)
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: fail (error) — The dot product of F(r(t)) and r'(t) is t², not “t², t²”. The solution incorrectly lists two terms and does not compute the dot product properly.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: fail (error) 2026-09-28 — The dot product of F(r(t)) and r'(t) is t², not “t², t²”. The solution incorrectly lists two terms and does not compute the dot product properly.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The parametrization is correct, the dot product calculation is accurate, 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.