The vector field is .

Greens theorem :

If C be a positively oriented closed curve, and R be the region bounded by C, M and N are the partial derivatives on an open region then

Greens theorem can be applied because C is a closed curve, but the theorem is applicable for positively oriented curves.

This curve rotates in a clockwise direction, so negative should be added before a integral.

Using polar coordinates,  and circle radius is 2.

So 

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