PX275 - D3c - application of green's theorem

\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1

\iint_{R} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A = \iint_{R} 1 d A = A

where, $A$ is the enclosed area

case 1

\oint_{C} P d x + Q d y = \oint_{C} x d y = \iint_{R} d A = A

\oint_{C} - y d x = \iint_{R} d A = A

\iint_{R} (1 - (- 1)) d A = 2 A

\begin{matrix} 2 A = \iint_{R} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A = \oint P d x + Q d y \\ ∴ A = \frac{1}{2} \oint_{C} x d y - y d x \end{matrix}