  The volume cut off by a hyperplane from a bounded body with smooth boundary
in $R^{2k}$ never is an algebraic function on the space of hyperplanes: for k=1
it is the famous lemma XXVIII from Newton's Principia. Following an analogy of
these volume functions with the solutions of hyperbolic PDE's, we study the
local version of the same problem: can such a volume function coincide with an
algebraic one at least in some domains of the space of hyperplanes,
intersecting the body? We prove some homological and geometric obstructions to
this integrability property. Based on these restrictions, we find a family of
examples of such "locally integrable" bodies in Euclidean spaces
