Tried using y=0..1/4, x=1/2-y..1 and y=1/4..1, x=y..1 as integration limits and also tried the reverse order but there is always problem with integrating 1/y up to the x axis. Any help please? Thanks.

If you carefully examine R, I think you'll find that the x-axis doesn't even touch it. If you look here, you can see that R is the region enclosed by the triangle. So, using this information, can you write down the integral?

I thought R was the region between the line y = x and the x axis (by the first 2 conditions) and then to the right of the line y = .5-x. Not sure what I am misreading if it isn't. ..

I think you need to be absolutely sure which side of a line is in the allowed region. To do that, pick a point that is obviously on one side of a line, and plug in the coordinates of that point into the inequality, and see if it holds. Does this help?

Tried using y=0..1/4, x=1/2-y..1 and y=1/4..1, x=y..1 as integration limits and also tried the reverse order but there is always problem with integrating 1/y up to the x axis. Any help please? Thanks.