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Green's theorem converts a closed line integral into a double integral over the enclosed region: ∮(P dx + Q dy) = ∬(∂Q/∂x − ∂P/∂y) dA. When the integrand is constant it is simply that value times the area.
Green's theorem
∮C (P dx + Q dy) = ∬R (∂Q/∂x − ∂P/∂y) dA
The circulation is (3 − 1) × 12 = 24.
Choosing P = −y/2 and Q = x/2 makes the integrand 1, so the line integral equals the area — the principle behind the planimeter.