Answer
$4 \pi \sqrt 5$
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}^{2}(2y-1) \cdot \sqrt {1+4} dy$
or, $= 2\sqrt 5 \pi (y^2-y) _{1}^{2}$
Thus, $Surface \space Area=4 \pi \sqrt 5$
