What does integration of g(x)f(x)dx mean with known bounds? What does $\int_{0}^{a}g(x)f(x)dx$ mean if $f(x)$ is the probability density function of $X$?
I do know $\int_{-\infty}^{+\infty}xf(x)dx$ is the expected value of $X$, from Wikipedia. But I am not sure how to interpret when bounds are given.

$$
{\rm E}[g(X)\mathbf{1}_{0<X<a}]
$$
