Answer
$$\frac{\pi}{27}(145 \sqrt{145}-1) $$
Work Step by Step
Given
$$y=x^ 3, \ \ \ \ \ 0 \leq x \leq 2$$
Since
$$\frac{dy}{dx}= 3x^2$$
Then
\begin{aligned}
S&= 2 \pi \int_{0}^{2} y\sqrt{1+\left[\frac{dy}{d x}\right]^{2}} d x\\
S&=2 \pi \int_{0}^{2} x^{3} \sqrt{1+\left(3 x^{2}\right)^{2}} d x\\
S&=2 \pi \int_{0}^{2} x^{3} \sqrt{1+9 x^{4}} d x
\end{aligned}
Let
$$u=1+9x^4\ \ \ \ \ du=18xdx$$
and
$$x=0\ \to\ u=1,\ \ x=2 \to \ u=145 $$
Then
\begin{aligned}
S&=2 \pi \int_{1}^{145} \sqrt{u} \frac{d u}{36}\\
& =\frac{\pi}{18} \int_{1}^{145} u^{1 / 2} d u\\
&=\frac{\pi}{18}\left[\frac{2}{3} u^{3 / 2}\right]_{1}^{145}\\
&=\frac{\pi}{18}\left[\frac{2}{3}(145)^{3 / 2}-\frac{2}{3}\right]\\
&=\frac{\pi}{18} \cdot \frac{2}{3}[145 \sqrt{145}-1]\\
&=\frac{\pi}{27}(145 \sqrt{145}-1)
\end{aligned}
