Area with Green's theorem
Problem 12.217 · easy
Use Green's theorem to evaluate \( \displaystyle \oint_C \left(- x y^{2}\right) dx + \left(2 x\right) dy \), where \( \displaystyle C \) is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 3] \), oriented counterclockwise.
- \[ \frac{d}{d x} 2 x - \frac{\partial}{\partial y} \left(- x y^{2}\right) = 2 x y + 2 \]∂Q/∂x − ∂P/∂y.✓ Proved
- \[ \int\limits_{0}^{3}\int\limits_{0}^{3} \left(2 x y + 2\right)\, dy\, dx = \frac{117}{2} \]∬_D (∂Q/∂x − ∂P/∂y) dA.✓ Proved
Answer \( \frac{117}{2} \)
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. The partial derivatives are calculated correctly, and the double integral over the specified rectangular region yields the correct result.
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. The partial derivatives are calculated correctly, and the double integral over the specified rectangular region yields the correct result.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The partial derivatives are calculated incorrectly. The derivative of -xy^2 with respect to y is -2xy, so the integrand should be 2 - (-2xy) = 2 + 2xy, but the solution claims the derivative of -xy^2 with respect to y is -2xy (implied by the result 2xy+2) yet writes the term as 2xy+2. Wait, let's re-read carefully. P = -xy^2, Q = 2x. dQ/dx = 2. dP/dy = -2xy. dQ/dx - dP/dy = 2 - (-2xy) = 2 + 2xy. The equation says 'Derivative(2*x, x) - Derivative(-x*y**2, y) = 2*x*y + 2'. This is algebraically correct (2 - (-2xy) = 2+2xy). However, the label says '∂Q/∂x − ∂P/∂y'. This is correct. Let's check the integral. Integral of (2+2xy) dy from 0 to 3 is [2y + xy^2] from 0 to 3 = 6 + 9x. Then integrate (6+9x) dx from 0 to 3 is [6x + 4.5x^2] from 0 to 3 = 18 + 4.5(9) = 18 + 40.5 = 58.5 = 117/2. The calculation is actually correct. Why did I think it was wrong? Let me re-read the prompt's instruction. 'Derivative(2*x, x) - Derivative(-x*y**2, y) = 2*x*y + 2'. d/dx(2x) = 2. d/dy(-xy^2) = -2xy. 2 - (-2xy) = 2 + 2xy. The equation is correct. The integral is correct. The answer is correct. Is there a subtle error? The problem asks to use Green's Theorem. The solution sets up the double integral of (dQ/dx - dP/dy). This is correct. The bounds are correct. The orientation is counterclockwise, which is standard for Green's Theorem. The result is correct. Wait, looking at the first line again. 'Derivative(2*x, x) - Derivative(-x*y**2, y) = 2*x*y + 2'. This is mathematically true. But is the notation misleading? No. Is there a sign error in the problem statement vs solution? P = -xy^2, Q = 2x. Green's: integral P dx + Q dy = double integral (dQ/dx - dP/dy) dA. dQ/dx = 2. dP/dy = -2xy. Difference = 2 + 2xy. Correct. Integral of 2+2xy over [0,3]x[0,3]. Inner integral dy: 2y + xy^2 evaluated 0 to 3 -> 6 + 9x. Outer integral dx: 6x + 9/2 x^2 evaluated 0 to 3 -> 18 + 81/2 = 36/2 + 81/2 = 117/2. Everything seems correct. Why would it fail? Maybe the 'unchecked' tag implies I shouldn't check the algebra? 'Each line... is either an EQUATION... do not re-check the algebra... or a SENTENCE... The sentences are what you are here for.' There are no sentences in the provided solution text, only equations. The prompt says 'Review it.' If there are no sentences, do I pass? Or is the lack of explanation a style issue? Or is there a hidden error? Let's look closer. 'Derivative(2*x, x) - Derivative(-x*y**2, y) = 2*x*y + 2'. This is an equation. 'Integral(...) = 117/2'. This is an equation. There are no sentences. The instructions say 'Check that... each conclusion follows from the lines before it'. The conclusion (the answer) follows from the integral. The integral follows from the derivative. The setup is implicit in the equations. Is the setup modeled correctly? Yes. Is the theorem applied correctly? Yes. Is the result correct? Yes. Perhaps the 'severity' is 'style' because there is no text explaining the steps? Or 'pass'? Let's look for a trap. Did I miss a negative sign? P = -xy^2. dP/dy = -2xy. dQ/dx - dP/dy = 2 - (-2xy) = 2+2xy. Correct. Bounds 0 to 3. Correct. Orientation CCW. Correct. Result 117/2. Correct. I will mark it as pass, but note the lack of explanatory text if that's considered a style issue. However, usually 'pass' is for sound solutions. This is sound. But wait, is the first line actually an equation? It has a comment '-- ∂Q/∂x − ∂P/∂y'. This is part of the line. It's an equation with a comment. It's not a sentence. I will pass it.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.