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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.

Questions:

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

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Unless you’re teaching at an elite school or an honors section $-$ and the mention of Stewart suggests that you’re probably not $-$ I think that trying seriously to develop the theory is just asking for great disappointment. –  Brian M. Scott Mar 30 '13 at 3:07
    
Is there a way you can give us a sense of the "level" of the class? Is it the highest-level class taken by freshman at your institution? Is there some well-known class you can compare it to, like math 55 at Harvard or 207 at Chicago? –  Alex Becker Mar 30 '13 at 3:16
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If your students were to be next year in one of my classes, I would ask, please don't do it. And I suspect that instructors in Engineering, Physics, Chemistry, $\dots$ would be even more vehement. –  André Nicolas Mar 30 '13 at 3:50
    
I do agree that Stewart's book is quite terrible, and Spivak's is miles ahead in terms of actual substance, but using baby Rudin in a freshman course is pushing it a bit, I think. –  noobProgrammer Mar 30 '13 at 14:41
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3 Answers

General advice for freshman-level course: Follow the textbook. Don't add variants of your own.

You will choose your own textbook? If so, presumably you have experience teaching similar students at that institution. If not, ask someone who has such experience. The question should not be whether it "does it" for you. But whether it is good for the students. These may not be the same thing...

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Would upvote twice if I could. –  Brian M. Scott Mar 30 '13 at 3:13
    
@BrianM.Scott, I'd do it for you, but I also have one vote only :-( –  vonbrand Mar 30 '13 at 4:51
    
Go for the sloppiness. Emphasise intuition, geometry, visualization, and applications. Unless these folks are going to become research mathematicians, they won't care about proving anything. Plausibility is enough. –  bubba Mar 30 '13 at 6:58
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Riemann integration is good for numerical approximation (right-hand rule, left-hand rule, midpoint, trapezoidal). It lets students get their hands on examples more. Darboux integrals aren't really easy to calculate unless the function is monotonic.

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Part of the point of college is to learn the craft, not just the subject. But of all the things you could give the students by a single deviation from the textbook, why Darboux integrals? Not infinite series, recognizing measures in an integral, or linear maps? Each of those would greatly clarify foundations, allowing students to remake calculus after their intuitions.

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