If we set $\mathscr{B}:=\{\alpha\in A\mid\text{$\alpha$ is integral over $\mathscr{A}$}\}$, then

$$\mathscr{B}=\bigcap_{x\in\operatorname{Spec}(A)^{\mathrm{an}}_{\mathscr{A}}}(A,|\raisebox{1.72218pt}{.}|_{x})_{\leq 1},$$

where $(A,|\raisebox{1.72218pt}{.}|_{x})_{\leq 1}:=\{\alpha\in A\mid|\alpha|_{x}\leq 1\}$.