Answer
$\dfrac{98 \pi}{81}$
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, $S =\int_{0}^{2} 2 \pi \dfrac{x^3}{9} \sqrt {1+(\dfrac{x}{3})^2} dx= \int_0^2 x^3 (\dfrac{2 \pi}{27}) \sqrt {x^4+9} dx$
Suppose $a =x^4+9$ and $da=4x^3 dx$
$ \dfrac{2 \pi}{27} \times \int_0^2 \dfrac{\sqrt a da}{4}= \dfrac{\pi}{54}[ (\dfrac{2}{3}) a^{(3/2)} ]_0^2=\dfrac{98 \pi}{81}$
