\begin{align*} \sigma^2 & = \int_{-1}^1 x^2 \cdot \frac{3}{4} \cdot (1-x^2) dx - \mu^2\\ & = \int_{-1}^1 \cdot \frac{3}{4} \cdot (x^2-x^4) dx - 0\\ & = \frac{3}{4} \cdot (x^3 /3 -x^5 / 5) \big |_{-1}^1\\ & = \frac{3}{4} \cdot 2 \cdot (1/3-1/5)\\ & = \frac{3}{4} \cdot \frac{4}{15}\\ & = \frac{1}{5} \end{align*}