Area with Green's theorem
Problem 12.211 · easy
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(0\right) dx + \left(x^{3}\right) dy \), where \( \displaystyle C \) is the triangle with vertices \( \displaystyle (0, 0) \), \( \displaystyle (3, 0) \), \( \displaystyle (0, 3) \), oriented counterclockwise.
- \[ - \frac{d}{d y} 0 + \frac{d}{d x} x^{3} = 3 x^{2} \]∂Q/∂x − ∂P/∂y.✓ Proved
- \[ \int\limits_{0}^{3}\int\limits_{0}^{3 - x} 3 x^{2}\, dy\, dx = \frac{81}{4} \]∬_D (∂Q/∂x − ∂P/∂y) dA.✓ Proved
Answer \( \frac{81}{4} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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, identifying the integrand as 3x^2 and setting up the double integral over the triangular region with correct bounds.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies Green's Theorem, identifying the integrand as 3x^2 and setting up the double integral over the triangular region with correct bounds.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies Green's Theorem, identifying the integrand as 3x^2 and setting up the double integral over the triangular region with correct bounds. The final result is correct.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.