  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, assuming
box-valued input uncertainties, we establish that this compact convex reach 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.
