  The integration of the exponential of the square of the moment map of the
circle action is studied by a direct stationary phase computation and by
applying the Duistermaat-Heckman formula. Both methods yield two distinct
formulas expressing the integral in terms of contributions from the critical
set of the square of the moment map. The cohomological pairings on the
symplectic quotient, including its volume (which was known to be a piecewise
polynomial), are computed explicitly using the asymptotic behavior of the two
formulas.
