While I see your point in general, I think that in this specific case the original poster gave up on the analytical approach too quickly, and came up with a solution that I don't think does what he thinks it does. (As I said, this particular problem didn't involve difficult, advanced mathematics -- it involved being able to solve a quadratic equation and plug the results into a black box pulled off the mathworld site)

Among other things, the original solution is not symmetric with respect to the distributions: if I reverse distribution 1 and distribution 2, I should get the same answer, right? If not, I can hardly claim to be measuring something related to the union of two distributions. Also, I wonder whether measuring the area under two curves is in fact the appropriate thing to do given the problem.