Answer
$$0$$
Work Step by Step
Stoke's Theorem states that $\oint F \cdot dr=\iint _S (\nabla \times F) \cdot n d\sigma $
$$ n=\dfrac{2}{7}i+\dfrac{6}{7}j-\dfrac{3}{7}k \\ d \sigma=\dfrac{7dA}{3}$$
Now, $$\iint _S (\nabla \times F) \cdot n d\sigma=\iint _{R} (\dfrac{6}{7} y)(\dfrac{7dA}{3}) \\ =\int_0^{2 \pi} \int_0^{1} 2r \space \sin \theta \space r \space dr \space d \space \theta \\=\int_0^{2 \pi} \dfrac{2}{3} \space \sin \theta \space \space d \theta \\ =\dfrac{2}{3}(0-0) \\=0$$
