If we have that $X\sim f$, then $$f_{X|R_{i}}(x)=\begin{cases} \frac{f(x)}{P(X\in R_{i})} & \text{if }x\in R_{i} \\ 0 & \text{otherwise} \end{cases}$$so $$E[X|X\in R_{i}]=\int\limits _{-\infty}^{\infty}xf_{X|R_{i}}(x) \, dx =\frac{ \int\limits _{R_{i}}xf(x) \, dx }{\int\limits_{R_{i}} f(x) \, dx }=\frac{ \int\limits _{R_{i}}xf(x) \, dx }{P(X\in R_{i})}$$