NOTE: Other threads suggest solving it with Gauss' Law. I'd like to see an approach through direct integration, no full followthrough necessary.. 1. The problem statement, all variables and given/known data Consider a sphere with a uniform distribution of charge ρ (ro). Inside the sphere is a cavity (spherical). Calculate the electric field at any given point of the cavity (vacuum). Radius of sphere: b radius of cavity (spherical): a a<b Distance from center of sphere to center of cavity is vector c. Please note the cavity is not necessarily centered. 2. Relevant equations Answer to the question is E=(ρc)/(3ε) ε is permissiveness of vacuum. 3. The attempt at a solution So my initial approach to this would be through direct integration. I'm not sure if Gauss' Law is applicable here, if it is great, that's one way to do the problem though I'd still like to see it done through direct integration if it is not too much of a hassle. So far the only problems I've seen done with direct integration are uniformly charged non conducting strings of varying lengths (infinite or length L). My understanding of these problems is you find some sort of way to represent the differential of charge along the volume of sphere dq. I believe in this case you'd have to use a double/triple integral when using this method, something I really am not sure how to apply in this case. I do have a solid understanding of vector calculus, I just need to get a foothold on how to start seeing differentials in physics. I only hope that when I say I have no idea how to even begin, you will understand I really want to make the effort to do this, but "I can't even", as the youth of today says. It'd be great to see tips on how you guys picture it, some pre-solving ritual or high quality online material that could help me out. Thank you in advance.