Answer
$\dfrac{12 \pi}{5}$
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}^{1} (2) \dfrac{(1-x^{2/3})^{3/2}}{x^{1/3}} dx $
Suppose $a =1- x^{2/3} \implies da= \dfrac{-2}{3x^{1/3}} dx$
or, $=-6 \pi \int_{0}^{1} a^{3/2} da$
or, $= -6 \pi [\dfrac{2a^{5/2}}{5}]_0^1$
or, $=\dfrac{12 \pi}{5}$
