The Riemann integral has a good geometrically motivated definition. So one teaches it but disadvantages are that even in elementary analysis it is not a complete tool. Lebesgue integral has the advantage that it is defined in ageneral set up and can handle multiple integration very well. the book by Apslund and Bungart used Mikusinski's definition and one can define integral quite quickly .It is reasonably intuitive but not as intuitive as Riemann.the approach does not require measure theory Daniel's approach does not require to define measure but again the class of upper functions and extension process is not intutive. these three definitions are not constructive but Riemann integral has constructive definition.
But the gauge integral (Henstock-Kurzweil) covered nicely in C. Swartz book is intuitive and a nice generalization of Riemann. Every derivative is integrable. There are no improper integrals . All improperly integrable functions are gauge integrable as well. the space L! is precisely the space of absolutely integrable mappings.the proof of fundamental theorem of calculus ( part Ii) is beautiful and elegant. We just use a gauge rather than mesh of a partition or a NET based o refinements.The integral is super Lebesgue yet definition is almost verbatim similar to Riemann integrable We can have a nice uniform convergence theorem. For graduate courses one can have The usual dominated convergence theorem( slightly more general version than for Lebesgue integral). Further the differential calculus in Banach spaces can be carried out without any restrictive assumption like continuity of derivative and we can have nice integral version of mean value theorem . the version is same as in along (analysis 1) but better as Lang restricts to Cauchy integral. On real line the integral is the perfect integral a satisfactory theory exists on euclidean spaces as well . one can define measure using this integral . about generality there is a question which i have asked on this forum. Professor buck in his book "Garden of integrals ) Terms this integral as integral of 21 st century.
