Work done to construct Dielectric Sphere with Offset Hollow Cavity?

[TwoAyoyoprogrammer]

Homework Statement

Find the work done to assemble positively charged particles from infinitely far way, to create a uniformly charged dielectric sphere with charge +Q and radius R. There is an offset hollow cavity with radius R/2, and electrically neutral (no charge).

Relevant Equations

Work = integral dWork

My thinking would be to do a work integral
Work = integral dWork
= integral delta V dq
= integral delta V 4(pi r^2) dr

The problem is, is this possible with a single integral?
Due to the offset cavity, the electric field E will not be constant at a given r.

 

[Alex Schaller]

A picture is worth a thousand words... Would you sketch such a picture?