Area with Green's theorem
Problem 12.303 · easy
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(x y\right) dx + \left(x y^{2}\right) dy \), where \( \displaystyle C \) is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), oriented counterclockwise.
- \[ - \frac{\partial}{\partial y} x y + \frac{\partial}{\partial x} x y^{2} = - x + y^{2} \]∂Q/∂x − ∂P/∂y.✓ Proved
- \[ \int\limits_{0}^{2}\int\limits_{0}^{2} \left(- x + y^{2}\right)\, dy\, dx = \frac{4}{3} \]∬_D (∂Q/∂x − ∂P/∂y) dA.✓ Proved
Answer \( \frac{4}{3} \)
Every line of this solution was proved by the computer algebra system SymPy. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ 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 | the line integral around the boundary, computed directly |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies Green's Theorem, computing the partial derivatives and the double integral over the specified rectangular region with correct bounds and orientation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies Green's Theorem, computing the partial derivatives and the double integral over the specified rectangular region with correct bounds and orientation.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The setup in line 1 calculates ∂Q/∂x - ∂P/∂y as y^2 - x, but labels it as '-x + y**2' which is correct algebraically, however the comment says '∂Q/∂x − ∂P/∂y' while the code shows 'Derivative(x*y, y) + Derivative(x*y**2, x)'. Derivative(x*y, y) is ∂P/∂y = x. So the code computes x + y^2. This is incorrect. Green's theorem requires ∂Q/∂x - ∂P/∂y. ∂Q/∂x = y^2. ∂P/∂y = x. So the integrand should be y^2 - x. The code computes x + y^2. This is a sign error in the setup of the double integral.gpt-oss:20b: pass 2026-10-12
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/greens_area, checked 2026-10-12 with SymPy 1.14.0.