An analytic expression (i.e., a formula) is probably not possible for an arbitrary cone and sphere.
I'm sure it is possible. I'll try to get this rigorous in a few hours when I have time - but in summary, you find the equations for 'membership' in each volume (the sphere, the cone), you rearrange them algebraically until they are expressed in terms of integrable parameters. For example:
-use the axis of the cone (parameter
a or something)
-at arbitrary a, consider the disk at a bounded by the cone (in other words, the flat circle inside the cone orthogonal to its axis)
-extend this plane to infinity: all such planes have simple form ax+by+cz=d, where all a,b,c are fixed and depend on the cone's axis
-get the equation of the intersection of this plane with the sphere (it's either nothing, or a perfect circle [or a point, but that has no area...])
-get the equations of the area in the intersection of this circle, and the cone (two circles in the same plane - I'd probably first find the arbitrary formula for two circles, radius r1, r2, distance d apart - its not too hard to find) [special case to watch out for - one circle is inside the other]
-repeat, integrating over the axis of the cone ("a")
The integral should be reasonably analytical, if you integrate over precisely that length of the axis along which intersection occurs. Or if it doesn't work, first split the problem into the few possible kinds of intersection, identity the regions, and treat each case individually. I'll revisit this tonight.
-rachmaninoff
