  We develop some useful techinques for integrating over Higgs branches in
supersymmetric theories with 4 and 8 supercharges. In particular, we define a
regularized volume for hyperkahler quotients. We evaluate this volume for
certain ALE and ALF spaces in terms of the hyperkahler periods. We also reduce
these volumes for a large class of hyperkahler quotients to simpler integrals.
These quotients include complex coadjoint orbits, instanton moduli spaces on
R^4 and ALE manifolds, Hitchin spaces, and moduli spaces of parabolic Higgs
bundles on Riemann surfaces. In the case of Hitchin spaces the evaluation of
the volume reduces to a summation over solutions of Bethe Ansatz equations for
the non-linear Schroedinger system. We discuss some applications of our
results.
