If $\displaystyle \int_{a}^{b} x^{3} d x=0$ and $\displaystyle \int_{a}^{b} x^{2} d x=\frac{2}{3}$, then find the values of a and $\displaystyle b$.

$\displaystyle \int_{a}^{b} x^{3}\,dx = 0 \Rightarrow \dfrac{b^{4}-a^{4}}{4} = 0$$\displaystyle \Rightarrow b^{4}-a^{4} = 0$$\displaystyle \Rightarrow a = -b \quad (a \neq b)$$\displaystyle \int_{a}^{b} x^{2}\,dx = \dfrac{2}{3} \Rightarrow \dfrac{b^{3}-a^{3}}{3} = \dfrac{2}{3}$$\displaystyle \Rightarrow b^{3}-a^{3} = 2$$\displaystyle \Rightarrow b^{3} = 1$$\displaystyle \Rightarrow b = 1$$\displaystyle \Rightarrow b = 1,\ a = -1$