Answer
$\dfrac{\sqrt {21}}{3}$
Work Step by Step
When the surface is orientable then the flux through a surface is defined as: $\iint_S F \cdot dS=\iint_S F \cdot n dS$
where, $n$ is the unit vector.
Consider $\iint_S x dS =\iiint_{D}x \sqrt {(-4)^2+(2)^2+1} dA$
or, $\iint_S x dS=\sqrt{21} \iint_D dA= \int_{0}^{1} \int_{2x-2}^0 x dy dx \times \sqrt{21}= \int_{0}^{1} [xy]_{2x-2}^0 x dx \times \sqrt{21}$
$\iint_S x dS =- \int_{0}^{1} 2x^2-2x dx \times \sqrt{21}=- \sqrt{21} [(\dfrac{2x^3}{3})-x^2]_{0}^{1}=-\sqrt {21} \times \dfrac{2(1)^3}{3}-(1)^2]_0^1=\dfrac{\sqrt {21}}{3}$
