Jim writes: << [W]hy would it make sense pedagogically to use sub-intervals of unequal widths as part of the _definition_ of the integral, and then state a _theorem_ that says that if a function is integrable then you might as well use equal-width sub-intervals (which is what Stewart does), rather than _define_ the integral using equal-width sub-intervals, and then state a _theorem_ that says that you can use more general partitions and still get Riemann sums that converge on the correct value, as long as the mesh goes to zero?
Seems to me that to define the integral in terms of the same limit for a wide variety of partitions would remove any suspicion that the nice way it turns out is solely an artifice of using equal-length intervals. Then to specialize to equal-length partitions when *calculating* the integral makes sense, just for the sake of convenience. This strategy, however, would certainly benefit from a theorem in between the above definition of integrals and the calculation thereof, stating that a large class of functions (e.g., piecewise continuous ones) actually *are* integrable in the sense mentioned. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele