Answer
$ \dfrac{(2 \sqrt 2-1)\pi }{9}$
Work Step by Step
Integrate the integral to calculate the surface area as follows:
Surface area, $S=2\pi \int_{c}^{d}(y) \sqrt {1+(\dfrac{dy}{dx})^2}$
or, $ =2 \pi \int_{0}^{1} \dfrac{x^3}{3} \sqrt {1+x^4} dx$
or, $=(\pi/6)\times \dfrac{2}{3} [(1+x^4)^{3/2} ]_0^1$
or, $= \dfrac{(2 \sqrt 2-1)\pi }{9}$
