I strongly prefer Darboux's method to the one commonly found in introductory level calculus texts such as Stewart, but I'm worried that it might be a bit overwhelming for my freshman level calculus class. My aim is to develop the theory, proving all of the results we need such as FTC, substitution rule, etc. If I can get everyone to buy into the concepts of lub and glb, this should be a fairly neat process. But that's potentially a big "if". Even worse, maybe they just won't care about the theory since they know I will not ask them to prove anything in an assignment.
It seems to me that there is very little middle ground here. You have to present integration the right way, paying attention to all the details, or accept a decent amount of sloppiness. In either case the class will likely lose interest.
Would you attempt the Darboux method? I would be using Spivak's Calculus / Rudin's Real Analysis as guides. I suppose there's no way of dumbing this down.
Otherwise, could you recommend a good source for the standard Riemann integral? Stewart just doesn't do it for me. Thanks