Answer
$\dfrac{21 \sqrt {21} -17 \sqrt {17}}{12}$
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 and $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$
Consider $\iint_S x dS =\iint_{R} x \times \sqrt {17 +4x^2} dA=\iint_{R} x \times \sqrt {17 +4x^2} dx dz$
and $\iint_S x dS=\int_{0}^{1} dz \times \int_0^1 \sqrt {17 +4x^2} dx =\int_{0}^{1} x \sqrt {17 +4x^2} dx $
Suppose $17+4x^2 =t; dt=8x dx$
$\iint_S x dS= \int_{17}^{21} \int_1^{17} \sqrt t \dfrac{dt}{8}=\dfrac{1}{8} \times [(\dfrac{2}{3}) t^{3/2}]_{17}^{21}$
and $\iint_S x dS=\dfrac{1}{12} [t (t^{1/2}]_{17}^{21}=(\dfrac{1}{12}) \times (21 \sqrt {21} -17 \sqrt {17})=\dfrac{21 \sqrt {21} -17 \sqrt {17}}{12}$
