  Using the Berline-Vergne integration formula for equivariant cohomology for
torus actions, we prove that integrals over Grassmannians (classical,
Lagrangian or orthogonal ones) of characteristic classes of the tautological
bundle, can be expressed as iterated residues at infinity of some holomorphic
functions of several variables. The results obtained for these cases, can be
expressed as special cases of one formula, involving the Weyl group action on
the characters of the natural representation of the torus.
