Answer
$\dfrac{21 \sqrt {21} -17 \sqrt {17}}{12}$
Work Step by Step
Here, we have $\iint_S x dS =\iint_{R} x \times \sqrt {17 +4x^2} dA$
$=\iint_{R} x \times \sqrt {17 +4x^2} dx dz$
$=\int_{0}^{1} dz \int_0^1 \sqrt {17 +4x^2} dx $
$=\int_{0}^{1} x \sqrt {17 +4x^2} dx $
Plug $a=17+4x^2 \implies da=8r dr$
$= \int_{17}^{21} \int_1^{17} a^{1/2} \times \dfrac{da}{8}$
$=\dfrac{1}{8} [(2/3) a^{3/2} ]_{17}^{21}$
$=\dfrac{1}{12} [a \sqrt a]_{17}^{21}$
$=(1/12) (21 \sqrt {21} -17 \sqrt {17})$
$=\dfrac{21 \sqrt {21} -17 \sqrt {17}}{12}$
