Editing Integral (section)

=== Formalization ===
While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of [[Rigor#Mathematical rigour|rigour]]. [[George Berkeley|Bishop Berkeley]] memorably attacked the vanishing increments used by Newton, calling them "[[The Analyst#Content|ghosts of departed quantities]]".<ref>{{harvnb|Katz|2009|pp=628–629}}.</ref> Calculus acquired a firmer footing with the development of [[Limit (mathematics)|limits]]. Integration was first rigorously formalized, using limits, by [[Bernhard Riemann|Riemann]].<ref>{{harvnb|Katz|2009|p=785}}.</ref> Although all bounded [[piecewise]] continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of [[Fourier analysis]]—to which Riemann's definition does not apply, and [[Henri Lebesgue|Lebesgue]] formulated a [[#Lebesgue integral|different definition of integral]], founded in [[Measure (mathematics)|measure theory]] (a subfield of [[real analysis]]). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the [[standard part]] of an infinite Riemann sum, based on the [[hyperreal number]] system.