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Simpl0S

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I finished working through Spivak's Calculus 3rd edition chapters 13 "Integrals", and 14 "The Fundamental Theorem of Calculus". By that I mean that I read the chapters, actively tried to prove every lemma, theorem and corollary before looking at Spivak's proofs, took notes into my notebook, and memorized the definitions, lemmas, theorems and corollary. After that I went on to solve some problems from chapter 13. I supplement exercises from Spivak's Calculus with Stewart's Calculus 7th edition because I find that Spivak lacks in mechanistic/computational exercises and Stewart includes mainly types of those exercises.

Up until now everything was working out good but I came to notice that the definition on integrals from Spivak and Stewart differ.

**Spivak's Definition:**

**Stewart's Definition:**Extracting information from this thread, from post #7:

https://www.physicsforums.com/threads/is-pure-math-useless.284355/#post-2031329

I came to notice that Spivak's Definition is called the darboux definition (Correct me if I am mistaken). I kinda hate this part of Spivak that he is trying to avoid naming the theorems, properties, definitions... It makes it hard when trying to reference them.

Which one is the more accurate or more used or more practical definition? What are the strengths/weaknesses? I prefer Spivak's definition because it appeals to me more intuitively after going through his inf's and sup's chapters and Spivak's definition is, to me, more beautiful.

I also have simply missed how to integrate. I think I have not been able to extract the big picture of integrals and the connection with the fundamental theorem(s) of calculus. Must I always use Spivak's definition with lower sums and upper sums when I want to integrate? How do I integrate? I had calculus in high school (it was called Analysis and it was non-rigorous, e.g. the teacher said: "Memorize the derivative rules and use them to find the derivative of the following function, then find maxima, minima, point of inflection"); in high school I remember that we used integrals in terms of what is called in German "Stammfunktion" which was the function F(x) = integral dx, and we used it to find the area under a curve and to find the area of a body which rotates across the horizontal axis or vertical axis. And again there we had some integration rules given by the teacher and we just used it in a computational way.

But now after reading Spivak I have no clue on how to integrate, for some reason. I might have missed or not completely understood the point of the chapters. Any help?