I've got a problem that I don't know how to solve. The question: Suppose that Y is a continuous random variable with density f(y) that is positive only if y> or equal to zero. If F(y) is the distribution function, show that

The book also gives a hint: If y > 0, and

Exchange the order of integration to obtain the desired result.

I've tried to change the integration order like this:

but how do I go on from there?

Thanks in advance.