Just to clarify: I'm well aware that for many problems, a non-uniform partition is handy. For instance, one can integrate x^c on [0,1] for any exponent c>0 as Fermat did, by dividing the domain [0,1] into infinitely-many sub-intervals of the form [q^(i-1),q^i] (with 0 < q < 1), which gives an infinite Riemann sum whose terms are in geometric progression, and then take the limit as q->1. (I'll guide my students through this in the case c=2 on their next homework assignment.) So, the issue isn't whether there's benefit from using non-uniform partitions; I already know that there is. The issue is whether one should (a) DEFINE the integral using uniform partitions and then PROVE (or at least state) a theorem about non-uniform partitions, or (b) DEFINE the integral using non-uniform partitions and then DERIVE (as a trivial consequence) a theorem about non-uniform partitions. Stewart takes course (b); it seems to me that (a) is easier for students to digest, and I wonder if there are reasons to prefer (b) that I might not be aware of. Jim Propp