Since you titled this "cylindrical coordinates, I do not understand why you are talking about "with respect to x and y". If you really wanted to use Cartesian coordinates, then you will need to divide the region between those two cylinders into non-overlapping portions where you can take y as a function of x or vice-versa.

I presume that you recognize that and are concentric cylinders of radii 2 and 3, repectively. when z= 0, the plane z= x+ y+ 5 becomes the line x+y= -5 which crosses the x and y axes at (0, -5) and (-5, 0). It's closest approach to the origin is where x= y: 2x= -5, x= y= -5/2. The distance from (-5/2, -5/2) to the origin is [tex]\sqrt{\frac{25}{4}+ \frac{25}{4}= \frac{5}{2}\sqrt{2}> 3. Whew! That means the top plane does not cut the lower planeinsidethe cylinders! That would have made the problem much harder!

So: In cylindrical coordinates, runs from 0 to , r runs from 2 to 3, and z runs from 0 to z= x+ y+ 5= . That is: