Answer
$$ 6$$
Work Step by Step
By implicit differentiation
\begin{aligned}
&\frac{2}{3} x^{-1 / 3}+\frac{2}{3} y^{-1 / 3} y^{\prime}=0\\
&y^{\prime}=-\frac{x^{-1 / 3}}{y^{-1 / 3}}=-\frac{y^{1 / 3}}{x^{1 / 3}}
\end{aligned}
and
\begin{aligned}
s=\int_{a}^{b} \sqrt{1+\left(y^{\prime}\right)^{2}} d x &=\int_{0}^{1} \sqrt{\frac{1}{x^{2 / 3}}} d x \\
&=\int_{0}^{1} \frac{1}{x^{1 / 3}} d x \\
&=\left[\frac{3}{2} x^{2 / 3}\right]_{0}^{1} \\
&=\frac{3}{2}
\end{aligned}
The total arc length is therefore $4 · \frac{3}{2} = 6.$
