Let $R$ be a ring (commutative noetherian with unit), and let $K(R)$ be its total ring of fractions (obtained by inverting all nonzerodivisors). Thus, $R \hookrightarrow K(R)$. Let $a \in K(R)$ be integral over $R$; in other words, $R[a]$ is a subring of $K(R)$ that is finite over $R$.
Is it necessarily true that $R[a]$ is flat over $R$? More to the point, since I'm expecting a negative answer: what is a concrete counterexample?