Practice Problem #13: Using Gauss's Law with a non-uniform charge density
Remarks:
Here we complicate life by asking you to work out the field for a spherical
distribution that is not uniform in some region. This means that you need
to explicitly integrate to get the charge enclosed. This is not really
so difficult but you need to keep track of "housekeeping" especially
when you work out the voltage.
This problem is just a little too hard to be on an hour exam. I might
put it on a final exam though.