  We consider the computation of the volume of the union of high-dimensional
geometric objects. While showing that this problem is #P-hard already for very
simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects
where one can: (1) test whether a given point lies inside the object, (2)
sample a point uniformly, (3) calculate the volume of the object in polynomial
time. All three oracles can be weak, that is, just approximate. This implies
that Klee's measure problem and the hypervolume indicator can be approximated
efficiently even though they are #P-hard and hence cannot be solved exactly in
time polynomial in the number of dimensions unless P=NP. Our algorithm also
allows to approximate efficiently the volume of the union of convex bodies
given by weak membership oracles.
  For the analogous problem of the intersection of high-dimensional geometric
objects we prove #P-hardness for boxes and show that there is no multiplicative
polynomial-time $2^{d^{1-\epsilon}}$-approximation for certain boxes unless
NP=BPP, but give a simple additive polynomial-time $\epsilon$-approximation.
