Utility function square root

Question

Could somebody tell my how to calculate $EU(X+2)$, where $U(X)={\sqrt X}$ and X is distributed evenly over the interval $[a,b]$?

Answer 1

$$EU(X)=E\sqrt {X+2}=\frac 1 {b-a} \int_a^{b} \sqrt {x+2} dx$$ $$=\frac 2 {3(b-a)} (x+2)^{1.5}|_a^{b}=\frac 2 {3(b-a)}[(b+2)^{1.5}-(a+2)^{1.5}]$$.