The author introduces an approach to analyse the construction of the integral closure of an affine ring. Instead of using the Noetherianness condition, a usual technique, here the approach applies to any algorithm that uses a class of extensions called divisorial. It estimates the number of steps by any method that progressively builds the closure by taking larger integral extensions. This setting gives quadratic multiplicity-based and dimension-independent bounds for the number of passes of the basic construction. The author also discusses an algorithm that does not use Jacobian ideals.