Without using a double integral, it is possible to find the volume by splitting it into two volumes which can be easily computed :
The intersection of the sphere and of the cone is given by
If , hence : the part of the cone such that is inside the sphere.
If , hence : the part of the sphere such that is inside the cone.
Thus the volume we're looking for is the sum of the volume bounded by the cone for (red surface, see attached images) and of the volume bounded by the sphere for . (green surface) Can you take it from here ?