  This is the first of a two part paper investigating the geometry of the
integrator reach sets, and the applications thereof. In this Part I, we
establish that this compact convex set is semialgebraic, translated zonoid, and
not a spectrahedron. We derive the parametric as well as the implicit
representation of the boundary of this reach set. We also deduce the closed
form formula for the volume and diameter of this set, and discuss their scaling
with state dimension and time. We point out that these results may be utilized
in benchmarking the performance of the reach set over-approximation algorithms.
