a hollow non-conducting sphere of outer radius 3R and inner radius R has a uniform volume charge density x. what is the field inside a spherical cavity of radius R touching the outer an inner spheres?

First consider a SOLID sphere, everything else remaining the same. It can be easily shown using simple vector methods that the field throughout the cavity is ENTIRELY due to the field of the region of region R inside the inner surface of the non-conducting sphere. This is obviously ZERO because this region is charge free.

Therefore, ANSWER is ZERO  

consider a gaussian surface just inside the innner sphere.Since,q(inside) is zero.the net flux cutting across it zero.due to symmetry,the field at all points on the gaussian surface is equal and hence,field at all points on the surface is zero.similarly consider gaussian surfaces with radius <R,and we can thus prove that field anywhere inside is zero.