As far as I have seen, the majority of modern introductory real analysis texts introduce Darboux integrals, not Riemann integrals. Indeed, many do not even mention Riemann integrals as they are actually defined (with Riemann sums as opposed to Darboux sums). However, they call the Darboux integrals Riemann integrals. Does anyone know the history behind this? I can understand why they use Darboux - I find it much more natural and the convergence is simpler in some sense (and of course the two are equivalent). But why do they call them Riemann integrals? Is this another mathematical misappropriation of credit or was Riemann perhaps more involved with Darboux integrals (which themselves may be misnamed)?
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Riemann and Darboux integrals are equivalent, but the Riemann-Stieltjes and Darboux-Stieltjes integrals, which extend the previous definitions to integrate a funtion with respect to another function, are not equivalent. Therefore, even pedagogically, I think they should be distinguished and named separately.
There are other examples, such as "An introduction to Real Analysis" by Wade. I don't know the history of these definitions at all. Once the dust settles over partitions, we have just one concept of integral left. The term "Riemann integral" is entrenched in so much of the literature that not using it isn't an option. One could use the term "Darboux integral" alongside "Riemann integral", but most students taking Intro to Real Analysis are sufficiently confused already. Mentioning names for the sake of mentioning names isn't what a math textbook should be doing. That job is best left to books on history of mathematics.
If you feel bad for Darboux, be sure to give him credit for the theorem about intermediate values of the derivative. (Rudin proves the theorem, but attaches no name to it.)
On a similar note: If I had my way, there'd be no mention of Maclaurin in calculus textbooks. Input: the total time spent by calculus instructors explaining the Maclaurin-Taylor nomenclatorial conundrum to sleep-deprived engineering freshmen. Output: