Answer
$\dfrac{98 \pi}{81}$
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_{0}^{2}(\dfrac{x^3}{9}) \cdot \sqrt {1+(\dfrac{x}{3})^2} dx$
or, $= \dfrac{2 \pi}{27} \int_0^2 x^3 \sqrt {x^4+9} dx$
Let us consider $a =x^4+9 \implies da=4x^3 dx$
Now, $ \dfrac{2 \pi}{27} \int_0^2 \dfrac{\sqrt a da}{4}= \dfrac{\pi}{54}[ (2/3) a^{3/2} ]_0^2$
Thus, $Surface \space Area=\dfrac{98 \pi}{81}$
