Since S is integral over R, chains of prime ideals pull down from S into R. Thus if Q1 contains Q2 contains 0 in S, P1 contains P2 contains 0 in R. The latter is impossible, hence every nonzero prime ideal in S is maximal. By definition 9, S is a dedekind domain.
How do we know S is noetherian? Often S is a finite integral extension, whence S is a finitely generated R module, and is noetherian. A noetherian R module is a noetherian S module, hence S is a noetherian ring.
How do we know S is integrally closed? this is usually handled case by case. You must prove that the extension is integrally closed, and then it becomes a dedekind domain.