#### Answer

$ 4 \pi$

#### Work Step by Step

The formula to determine the surface area is as follows:
$S= \int_{m}^{n} 2 \pi y \sqrt {1+(\dfrac{dy}{dx})^2}$
Now, $S =2 \pi \times \int_{0}^{\sqrt 2} (x) \sqrt {1+x^2(x^2+2)} dx \\= 2 \pi [ \dfrac{x^4}{4}+\dfrac{x^2}{2}]_{0}^{\sqrt 2} \\=4 \pi$