Review of A garden of integrals , by Frank E . Burk ( MAA , 2007 ) Dedicated to the memory of Ralph Henstock ( 1923 - 2007 )

Riemann, Lebesgue, Denjoy, Henstock–Kurzweil, McShane, Feynman, Bochner. There are well over 100 named integrals. Why so many? Some are of historical interest and have been superseded by better, newer ones. The Harnack integral is subsumed by the Denjoy. Some are equivalent, as are McShane and Lebesgue in Rn, and Denjoy, Perron, Henstock–Kurzweil in R. Some are designed to work in special spaces: Feynman for path integrals, where the domain is the set of all continuous functions on [0, 1], Bochner for Banach space-valued functions. Others are designed to invert special derivatives such as the symmetric derivative or distributional derivative. And, we continue to keep the Riemann integral around because it is so easy to define. Can’t we have just one super-integral that does everything? The problem is that if an integral is defined to work, for example, on all Banach spaces then it will probably be unnecessarily complicated when restricted to a simple setting such as the real line. There is also a pedagogical issue. We need to build up mathematics from simple bits before we can define the most abstract structures (Bourbaki notwithstanding). So, it seems we will have to live with this plethora of integrals.