Why is it when we do an expected value for a continuous random variable like exponential or uniform, we initially start off with the integral from -infinity to positive infinity, but then go to the integral of 0 to some number(or infinity in an exponential case). For example:

Y is an exponential random variable with parameter lambda = 0.2, given the event A = [y

Question

Why is it when we do an expected value for a continuous random variable like exponential or uniform, we initially start off with the integral from -infinity to positive infinity, but then go to the integral of 0 to some number(or infinity in an exponential case). For example:

Y is an exponential random variable with parameter lambda = 0.2, given the event A = [y<2].

a) what is the conditional pdf, fy|a(y)
b)Find the conditional expected value E[Y|A]

I already solved the answer, but why are we setting the integral from 0 to 2 for the expected value in this case?