Random variable $X$ has uniform distribution on section $[0,2]$. What's the expected value of variable $Y=frac{X^{4}}{2}$

The expected value of a probability distribution, $g(x)$, is defined as $int_{mathbb{R}}x g(x) dx$. In your case, $g(x)$ is the distribution of $frac{X^4}{2}$, where $X in U(0, 2)$. Using the substitution $x = frac{u^4}{2}$, which is equivalent to the law of the unconscious statistician, the integral becomes $$int_{0}^{2} frac{u^4}{2} cdotfrac{1}{2-0} mathrm d u$$

Can you finish from here?

There's probably a shortcut somewhere, so you can find $mathbf{E}Y$ without the pdf, but I don't know it. So first, it's the easiest to find CDF of $Y$: $$ P(Y<y) = P(X^4<2y) = P(0<X<(2y)^{frac{1}{4}}) $$ Since $X$ is positive, and $X^4$ is a strictly increasing funciton, it's easy enough to find it. From that, take a derivative to get pdf. Now, follow @BrianTung's suggestion by using the definition of $mathbf{E}X$