Answer
$$\dfrac{-73 \pi}{6} $$
Work Step by Step
$F \cdot n =\dfrac{-x^2 }{\sqrt {x^2+y^2}}-\dfrac{y^2 }{\sqrt {x^2+y^2}} -x^2-y^2$
Now, $I=\iint_{S} F \cdot n \ dS=\iint_{R} [\dfrac{-x^2 }{\sqrt {x^2+y^2}}-\dfrac{y^2 }{\sqrt {x^2+y^2}} -x^2-y^2] \ dy \ dx \\=-\int_0^{2 \pi} \int_{1}^{2} [\dfrac{r^2 }{\sqrt {r^2}} \times \cos^2 \theta+\dfrac{r^2)}{(\sqrt {r^2}} \times \sin^2 \theta+r^2 \cos^2 \theta +r^2 \sin^2 \theta] (r \ dr) d \theta \\=-\int_0^{2 \pi} \int_{1}^{2} (r^2+r^3) r dr d \theta \\= -\int_0^{2 \pi} [\dfrac{1}{3} r^3 +\dfrac{1}{4} r^4]_1^2 \ d \theta \\=\dfrac{-73 \pi}{6} $
