Here is another example of a NON-Guass's law problem, a partial ring of charge. This is a combination of sticks and rings, the two classic cases where we might have to calculate something like an E-field or potential on some axis. In any case like this, we need to use the fundamentals: point charges, which we know how to do exactly. We break the problem into a bunch of little charges, find the small contribution, and then add them all up using an integral! Let's check it out, and I hope it helps.