\begin{align*}
\int_0^1\dif y\int_y^1\frac{y}{\sqrt{1+x^3}}\dif x
&=\int_0^1\dif x\int_0^x\frac{y}{\sqrt{1+x^3}}\dif y\\
&=\int_0^1\frac{1}{\sqrt{1+x^3}}\left[\frac{1}{2}y^2\right]_0^x\dif x=\frac{1}{2}\int_0^1\frac{x^2}{\sqrt{1+x^3}}\dif x\\
&=\left[\frac{1}{3}\sqrt{1+x^3}\right]_0^1=\frac{1}{3}(\sqrt{2}-1)
\end{align*}