Expected value of a function vs average value

Question

I don't understand the different between these two. For example, the average value of $\sin ^{2}\theta$ should be $\frac{1}{2}$ on $x∈[0,2\pi]$ according to the mean formula $${\bar {f}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx.$$

However, WolframAlpha suggests that the expected value, $\operatorname {E} (\sin^2{x})=0.432...$

I don't understand what the different between these two is.

Answer 1

I think I got it.

You seem to evaluate this expresion: Expectation Sin[x]^2 but it's related to normal distribution of $x$. And the formula from your question is related to uniformly distributed $x$.