Moments of non integer order of uniform distribution

Question

Suppose that $U$ is a random variable from a uniform distribution on $[a, b]$.

Then, we can obtain the moment generating function of $U$, and by using that, we can get the $n$th order moment of $U$ for a natural number $n$.

But, how can I get the moments of $U$ such as the mean of $U^{2.7}$? That is, $r$th order moment for a real number $r$.

Answer 1

Assuming $[a,b]\subset[0,+\infty)$, for every $\alpha>-1$, $$E(U^\alpha)=\int_a^bu^\alpha\,\frac{\mathrm du}{b-a}=\frac1{\alpha+1}\,\frac{b^{\alpha+1}-a^{\alpha+1}}{b-a}$$