$\begin{aligned} & \int_a^b x^3 d x=0 \text { and } \int_a^b x^2 d x=\frac{2}{3} \\ & \Rightarrow\left[\frac{x^4}{4}\right]_a^b=0 \text { and }\left[\frac{x^3}{3}\right]_a^b=\frac{2}{3} \\ & \Rightarrow \frac{b^4}{4}-\frac{a^4}{4}=0 \text { and } \frac{b^3}{3}-\frac{a^3}{3}=\frac{2}{3} \\ & \Rightarrow a^4=b^4 \text { and } b^3-a^3=2\end{aligned}$
which is satisfied by $\mathrm{a}=-1$ and $\mathrm{b}=1$