We evaluate several infinite families of multidimensional integrals which are generalizations or analogs of Euler's classical beta integral. We first evaluate a ^-analog of Selberg's beta integral. This integral is then used to prove the Macdonald-Morris conjectures for the affine root systems of types S{Ci) and S{Cj) and to give a new proof of these conjectures for SiBC;), S{B(), 5 (B/) V and S*(£/) •