Unless you are talking about the portion that lies above the 2nd quadrant in the xy plane the surface area is infinite.

I think that this is what you are looking for.

If so this will work

using the formula

From here the calculation of the integral can be done by letting y go from to and x from to

Or you can notice the region in the 2nd quadrant is a triangle with a base of and a height of

So finally we get