If X ranges from 0 to 1, and the bounds of Y are defined in terms of X, you should be able to determine what the bounds of Y are.
Wild. Must be different pre-reqs throughout the country. Calc 1 is pretty much it for most Stats classes at the Universities/Colleges here. It's not until you're deep in upper division Stat classes that a full compliment of Calc is required; definitely not for lower division stat courses.
This is basic calc 3.
YOU must draw the region.
First draw the lines y=x, y=2x and x=1.
Then find the three vertices, (0,0), (1,1) and (1,2).
This rectangle is easier to integrate via dydx than dxdy.
Once you have the bounds the three integrals are all the same except for the integrand.
IT's better to look at this two dimensionally and NOT obtain the marginals.
Obtaining the marginals forces you to do a particular order of integration
and I prefer dydx in all three cases in this example.
Likewise