In order to be able to solve these kinds of volume questions, you must learn to draw or at least visualize the area you are finding. I will help you here, but eventually you're going to have to be able to do it yourself.
First of all the z=0 bound is unnecessary as this elliptic paraboloids vertex is at (0,0,0). z is bounded above by your function. y goes from x to 1, and x goes from 0 to infinity.
EDIT: These bounds could also describe an area where y goes from x to 1 and x goes from -1 to 0, which may be what your professor is looking for since its supposed to be an enclosed volume