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Let $R$ be a region in the first quadrant of the $xy$ plane enclosed by a closed curve $C$ considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region $R$?

- $\displaystyle\iint_{R} dxdy$

- $\oint _{C}xdy$

- $\oint _{C}ydx$

- $\dfrac{1}{2}\oint _{C}\left ( xdy -ydx \right )$