Answer
$3 \pi \sqrt 5$
Work Step by Step
Integrate the integral to calculate the surface area as follows:
We have: $S_{A}= (2 \pi)\int_{a}^{b} y \sqrt {1+(\dfrac{dy}{dx})^2}$
or, $ =(2 \pi)\int_{1}^{3}(\dfrac{x}{2}+\dfrac{1}{2}) \cdot \sqrt {1+\dfrac{1}{4}} dy$
or, $= \sqrt 5 \pi (\dfrac{x^2}{4}+\dfrac{x^2}{2}) _{1}^{3}$
Thus, $Surface \space Area=3 \pi \sqrt 5$
