## Thomas' Calculus 13th Edition

$2 \times ( \ Area \ of \ the \ Square)$
The tangential form for Green Theorem's can be calculated as: Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy$ $\implies \oint_C xy^2 dx +(x^2y +2x) \ dy= \iint_{R} (\dfrac{\partial (x^2y+2x) }{\partial x}-\dfrac{\partial (xy^2) }{\partial y}) dx dy =\iint_{R} 2xy+2-2xy \ dx \ dy$ $\implies \oint_C xy^2 dx +(x^2y +2x) \ dy=2 \times ( \ Area \ of \ the \ Square)$