TITLE: Why are Darboux integrals called Riemann integrals?
QUESTION [21 upvotes]: 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)?

REPLY [7 votes]: 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.