On local rings of a normalization

Let $X$ an irreducible singular curve with a singular point $x$. Consider $A$, the normalization of $\mathcal{O}_{X,x}$, and $x_1,\dots,x_r$ the points over $x$ in the normalization of $X$. Why $A\cong \mathcal{O}_{x_1}\cap\dots\cap\mathcal{O}_{x_r}$? Moreover assume that the curve is over a field $k$. Why do the dimension of $A/\mathcal{O}_{X,x}$ is $r-1$?

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Here's a reduction: If $R$ is an integrally closed noetherian domain, then $R=\cap R_p$ where the intersection is over all primes $p\subset R$ that have height 1. Note $\mathcal{O}_x$ local and $A$ integrally closed, so why are $x_1, \ldots, x_r$ all the height 1 primes? – Matt Mar 4 '11 at 17:31