  Given an ideal in a polynomial ring, we show that the asymptotic resurgence
studied by Guardo, Harbourne, and Van Tuyl can be computed using integral
closures. As a consequence, the asymptotic resurgence of an ideal is the
maximum of finitely many ratios involving Waldschmidt-like constants (which we
call skew Waldschmidt constants) defined in terms of Rees valuations. We use
this to prove that the asymptotic resurgence coincides with the resurgence if
the ideal is normal (that is, all its powers are integrally closed).
  For a monomial ideal the skew Waldschmidt constants have an interpretation
involving the symbolic polyhedron defined by Cooper, Embree, H\`a, and Hoefel.
Using this intuition we provide several examples of squarefree monomial ideals
whose resurgence and asymptotic resurgence are different.
