Answer
$\dfrac{12 \pi}{5}$
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} =2 \pi \times \int_{0}^{1} (2) \dfrac{(1-x^{2/3})^{3/2}}{x^{1/3}} dx $
Let us consider $\space a =1- x^{2/3}$ and $da= \dfrac{-2}{3x^{1/3}} dx$
Now $S=-6 \pi \int_{0}^{1} a^{3/2} da \\= -6 \pi \times (\dfrac{2}{5})[a^{5/2}]_0^1 \\=\dfrac{12 \pi}{5}$