Answer
$0$
Work Step by Step
We need to work out with the line integral and evaluate the integrand of the double integral as follows:
$\int_{C} F \cdot dr= -\int_{C} \dfrac{y}{x^2+y^2} i+\int_{C} \dfrac{x}{x^2+y^2} j (dx i +dyj)=-\int_{C} \dfrac{y}{x^2+y^2} dx+\int_{C} \dfrac{x}{x^2+y^2} dy$
Green's Theorem states that:
$\oint_C M \,dx+ N \,dy=\iint_{D}(\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y})dA$
Now, $\oint_C-\dfrac{y}{x^2+y^2} dx+\int_{C} \dfrac{x}{x^2+y^2} dy=\iint_{D}(\dfrac{\partial (\dfrac{x}{x^2+y^2} )}{\partial x}-\dfrac{\partial (-\dfrac{y}{x^2+y^2} )}{\partial y})\ dA \\=\iint_{D} \dfrac{y^2-x^2}{(x^2+y^2)^2}-\dfrac{y^2-x^2}{(x^2+y^2)^2} \\ =0$
