  The theory of integration over R is rich with techniques as well as necessary
and sufficient conditions under which integration can be performed. Of the many
different types of integrals that have been developed since the days of Newton
and Leibniz, one relative newcomer is that of the Henstock integral, aka the
Henstock-Kurzweil integral, Generalized Riemann integral, or gauge integral,
which was discovered independently by Henstock and Kurzweil in the mid-1950s.
In this paper, we develop an alternative, sequential definition of the Henstock
integral over closed intervals in R that we denote as the Sequential Henstock
integral. We show its equivalence to the standard epsilon-delta definition of
the Henstock integral as well as to the Darboux definition and to a topological
definition of the Henstock integral. We then establish the basic properties and
fundamental theorems, including two convergence theorems, for the Sequential
Henstock integral and offer suggestions for further study.
