Cylindrical coordinates would be easiest. The first surface is the upper half of a hyperboloid of one sheet (see the attached graph). Because of the "hole" in the center of the graph, integrating first with respect to might make things difficult. Instead, choose the order The hyperboloid, in cylindrical form, is

and, solving for

This suggests the following limits:

So, the volume of the region is

I leave the integration to you.