  A polytope is integral if all of its vertices are lattice points. The
constant term of the Ehrhart polynomial of an integral polytope is known to be
1. In previous work, we showed that the coefficients of the Ehrhart polynomial
of a lattice-face polytope are volumes of projections of the polytope. We
generalize both results by introducing a notion of $k$-integral polytopes,
where 0-integral is equivalent to integral. We show that the Ehrhart polynomial
of a $k$-integral polytope $P$ has the properties that the coefficients in
degrees less than or equal to $k$ are determined by a projection of $P$, and
the coefficients in higher degrees are determined by slices of $P$. A key step
of the proof is that under certain generality conditions, the volume of a
polytope is equal to the sum of volumes of slices of the polytope.
