Area with Green's theorem
Problem 12.300 · easy
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(y^{2}\right) dx + \left(x^{3}\right) dy \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 9 \), oriented counterclockwise.
- \[ \frac{d}{d x} x^{3} - \frac{d}{d y} y^{2} = 3 x^{2} - 2 y \]∂Q/∂x − ∂P/∂y.✓ Proved
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3} r \left(3 r^{2} \cos^{2}{\left(\theta \right)} - 2 r \sin{\left(\theta \right)}\right)\, dr\, d\theta = \frac{243 \pi}{4} \]∬_D (∂Q/∂x − ∂P/∂y) dA in polar coordinates.✓ Proved
Answer \( \frac{243 \pi}{4} \)
✓ 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, computes the partial derivatives accurately, and sets up the polar integral with the correct Jacobian and limits. The final result is correct.
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, computes the partial derivatives accurately, and sets up the polar integral with the correct Jacobian and limits. The final result is correct.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies Green's Theorem, computing the integrand as 3x^2 - 2y. The conversion to polar coordinates and the evaluation of the double integral are correct, as the term involving sin(theta) integrates to zero over [0, 2pi], leaving only the contribution from 3r^3 cos^2(theta), which yields 243pi/4.
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.