Exam P Problem

[AlG]

An insurance policy is written to cover loss X where f(x)=(3/8)*x^2. The time (in hrs) to process a claim of size x, where 0<x<2, is uniformly distributed from x to 2x. Find P(T>3)?

I've seen two solutions to this problem that are nearly identical with the exception of the double integrals' boundaries. Variables are used for certain integration limits.

I don't see why a variable is needed in the final steps of integration. Why can't we simply use a double integral from 1.5 to 2 for dx & 3 to 4 for dt???

I'm not very savvy when it comes to integration and don't have a ton of practice with it, am I may be missing something obvious, my results don't yield .17

I get .328

[audia4]

The reason why you can't apply a double integral from 1.5-->2 and 3-->4 because as x is changing y is changing. Your solution will only be correct if the question were to say find P(T>3|X=(some value)), but this question is saying find P(T>3|1.5 <x<2), so as x changes the probability that T>3 will be a triangular region, ie the integral will be from 1.5-->2, and 3-->x.

I am not sure if I was clear, but I Hope this helps