Abstract: Bombieri and Selberg showed how Mehta's \cite{6 ,p.42} integral could be evaluated using Selberg's \cite{7} integral. Macdonald \cite{5, \S\S 5,6} conjectured two different generalizations of Mehta's integral formula. The first generalizations is in terms of finite Coxeter groups and depends on one parameter. The second generalization is in terms of root systems and the number of parameters in equal to the number of different root lengths. In the case of Weyl groups Macdonald showed how the first generalization follows from the second. We give a proof of the $\Cal I_3$ case of the first generalization and the $F_4$ case of the second generalization. As well we give a two parameter generalization for the dihedral group $\Cal H_2^{2n}$. The parameters are constant on each of the two orbits. We note that the $G_2$ case of the second generalization follows from our two-parameter version for $\Cal H_2^6$. Our proofs draw on ideas from Aomoto's \cite{1} proof of Selberg's integral and Zeilberger's \cite{10} proof of the $G_2^\vee$ case of the Macdonald Morris \cite{5, conj. 3.3} constant term root system conjecture. The problem is reduced to solving a system of linear equations. These equations were generated and solved by the computer algebra package MAPLE.