  In this paper, a geometric resolution of singularities algorithm is
developed. This method is elementary in its statement and proof, using explicit
coordinate systems as much as possible. Each coordinate change used in the
resolution procedure is one-to-one on its domain, and is of one of a few
explicit canonical forms. As applications of these methods to classical
analysis, two theorems are proven. First and foremost, a general theorem
regarding the existence of critical integrability exponents is established.
Secondly, a new proof of a well-known inequality of Lojasiewicz is given.
