Some of the above material is very technical. This is because it gives complete answers to the fundamental questions on existence of the integral and related theoretical considerations. However, most of the difficulties are artifacts. They should not even be considered! It was realized early in the twentieth century that these difficulties occur because, from the point of view of mathematics, this is not the right way to define an integral! Better results are obtained much more easily using the Lebesgue integral. Many of the technicalities related to Jordan content disappear almost magically when the right integral is used. However, the Lebesgue integral is more abstract than the Riemannn integral and it is not traditional to consider it in a beginning calculus course. If you are interested in the fundamental properties of the integral and the theory behind it, you should abandon the Riemannn integral which is an antiquated relic and begin to study the integral of the last century. An introduction to it is in [22]. Another very good source is [12]. This advanced calculus text does everything in terms of the Lebesgue integral and never bothers to struggle with the inferior Riemannn integral. A more general treatment is found in [18], [19], [23], and [20]. There is also a still more general integral called the generalized Riemannn integral. A recent book on this subject is [5]. It is far easier to define than the Lebesgue integral but the convergence theorems are much harder to prove. An introduction is also in [19].

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