Find the electric field magnitude in each region.

So far, this is what I have. Now I tried to make as a nice as I could but for some reason every time I attempted to save on a PDF file, I find some missing fonts. This one seems to be okay. But question is what am I to do with the rho outside of the cavity.

Inside the cavity the electric field is 0, no charge enclosed. between the cavity and the outer shell, integrate the charge density to the radius you are calculatina, i.e. some $r$. Outside the sphere, just integrate the entire shell minus the cavity to calculate the entire charge enclosed.