Before revision:  Thus, in order to translate the improved Fr\'echet--Hoeffding bounds into bounds on the expectation of f(S), we develop an alternative representation of multivariate integrals with respect to copulas that admits also quasi-copulas as integrators.
After revision:  Thus, in order to translate the improved Fr\'echet-Hoeffding bounds into bounds on the expectation of f(S), we develop an alternative representation of multivariate integrals with respect to copulas that admits also quasi-copulas as integrators.