\begin{align*} \sigma^2 & = E[X^2] - \mu^2\\ & = \int_{-1}^2 x^2 \cdot x^2/3 \; dx - \mu^2\\ & = \int_{-1}^2 x^4/3 \; dx - \left ( \frac{5}{4} \right )^2\\ & = \frac{2^5}{15} - \frac{(-1)^5}{15} - \frac{25}{16}\\ & = \frac{33}{15} - \frac{25}{16} = \frac{51}{80} \end{align*}