\begin{align*} \sigma^2 & = E[X^2] - \mu^2 = \int_a^b x^2 \frac{1}{b-a} dx - \mu^2\\ & = \frac{x^3}{3(b-a)} right |_a^b - \left ( \frac{a+b}{2} \right )^2\\ & = \frac{b^3-a^3}{3(b-a)} - \frac{a^2 + 2ab + b^2}{4}\\ & = \frac{4 b^2 + 4 ab + 4 a^2 - 3a^2 - 6 ab - 3b^2}{12}\\ & = \frac{b^2-2ab+a^2}{12} = \frac{(b-a)^2}{12}. \end{align*}