Answer
$4 \pi$
Work Step by Step
Our aim is to integrate the integral to compute the surface area. In order to solve the integral, we have:
$Surface \space Area(S_A)= (2 \pi)\int_{a}^{b} y \sqrt {1+(\dfrac{dy}{dx})^2}$
or, $ =(2 \pi)\int_{0}^{\sqrt 2} (x) \sqrt {1+x^2(x^2+2)} dx $
or, $ =(2 \pi)\int_{0}^{\sqrt 2} (x) (x^2+1) dx $
or, $= 2 \pi [ \dfrac{x^4}{4}+\dfrac{x^2}{2}]_{0}^{\sqrt 2}$
or, $=4 \pi$
