I'm confused about something that should be simple. Suppose I have a random uniform variable $X$ on $[0,1]$. It's fairly clear that the expected value of $X$ is 1/2. By integrating $x^2$ on $[0,1]$, I get that the expected value of $x^2$ is 1/3. I'm struggling to understand this intuitively, as I would expect it to be $1/4$ i.e. $(1/2)^2$.