Editing Riemann integral (section)

==Comparison with other theories of integration==
The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the [[Riemann–Stieltjes integral]], and most disappear with the [[Lebesgue integral]], though the latter does not have a satisfactory treatment of [[improper integrals]].  The [[gauge integral]] is a generalisation of the Lebesgue integral that is at the same time closer to the Riemann integral.   
These more general theories allow for the integration of more "jagged" or "highly oscillating" functions whose Riemann integral does not exist; but the theories give the same value as the Riemann integral when it does exist.

In educational settings, the [[Darboux integral]] offers a simpler definition that is easier to work with; it can be used to introduce the Riemann integral. The Darboux integral is defined whenever the Riemann integral is, and always gives the same result. Conversely, the [[gauge integral]] is a simple but more powerful generalization of the Riemann integral and has led some educators to advocate that it should replace the Riemann integral in introductory calculus courses.<ref>{{cite web|title=An Open Letter to Authors of Calculus Books|url=https://math.vanderbilt.edu/schectex/ccc/gauge/letter/|access-date=27 February 2014}}</ref>