\[\begin{align*} E(X) &= \int_a^b x \cdot\frac{1}{b-a}dx \\ &= \frac{1}{b-a} \int_a^b xdx \\ &= \frac{1}{b-a} \frac{x^2}{2} \bigg\rvert_a^b \\ &= \frac{1}{2(b-a)}(b^2-a^2) \\ &= \frac{1}{2(b-a)}(b-a)(b+a) \\ &= \frac{a+b}{2} \end{align*}\]
\[\begin{align*} E(X^2) &= \int_a^b x^2 \frac{1}{b-a}dx \\ &= \frac{1}{b-a} \int_a^b x^2 dx \\ &= \frac{1}{b-a} \frac{x^3}{3} \bigg\rvert_a^b \\ &= \frac{1}{3(b-a)}(b^3-a^3) \\ &= \frac{1}{3(b-a)}(b-a)(b^2+ab+a^2) \\ &= \frac{b^2+ab+a^2}{3} \end{align*}\]
\[\begin{align*} Var(X) &= E(X^2) - E(X)^2\\ &=\frac{b^2+ab+a^2}{3} - \left(\frac{a+b}{2} \right)^2 \\ &= \frac{b^2+ab+a^2}{3} - \frac{(a^2 + 2ab +b^2)}{4} \\ &= \frac{4b^2+4ab+4a^2 -3a^2 - 6ab - 3b^2}{12} \\ &= \frac{b^2-2ab+a^2}{12} \\ &= \frac{(b-a)^2}{12} \end{align*}\]
\[\begin{align*} M_X(t) &= \int_a^b e^{tx} \frac{1}{b-a} dx \\ &= \frac{1}{b-a} \int_a^b e^{tx}dx \\ &= \frac{1}{b-a} \left[\frac{1}{t} e^{tx} \right]_a^b \\ &= \frac{1}{t(b-a)}(e^{tb} - e^{ta}) \\ &= \frac{e^{bt} - e^{at}}{t(b-a)}, t \ne 0 \end{align*}\]