Question

Use Green’s Theorem to evaluate (Check the orientation of the curve before applying the theorem.)

Answer 1

Step 1:

The vector field is .

Green's 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

Green's 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

                

.

Solution :

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