Answer
$\dfrac{28 \pi}{9}$
Work Step by Step
$dS=\sqrt{(\dfrac{dx}{dt})^2+(\dfrac{dy}{dt})^2}=\sqrt{(\sqrt {t})^2+(t^{-1/2})^2}=\sqrt{\dfrac{t^2+1}{t}}$
This implies that
$S=\int_{0}^{\sqrt 3} (2 \pi) x ds$
or, $S=\int_{0}^{\sqrt 3} (2 \pi) (\dfrac{2}{3}t^{3/2}) (\sqrt{\dfrac{t^2+1}{t}}) dt=\dfrac{4 \pi}{3}\int_{0}^{\sqrt 3} t\sqrt{t^2+1} dt$
Plug $t^2+1=k \implies (2) dt=dk$
$S=(\dfrac{2 \pi}{3}) \int_{1}^{4} (\sqrt{k}) dk=\dfrac{28 \pi}{9}$
