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My math teacher was teaching us Riemann Sums a few days back, and how if you estimate the height of the rectangle to be at the upper/lower, as n approaches infinity, the area becomes exact.

But what I don't understand is why uppers/lowers matter at all. If I was to compute the Riemann sum of $F(x) = x^2$ in the bound $[0,5]$, I can do so rather easily with the upper limit of the rectangles. Why do I need to know the lower limit of the rectangles in the first place, then? To me, I feel like it's resulting in the same thing overall, so it's just another way to find things that's merely more complex.

Also, what about midpoint Riemann sums? If I already have 2 ways to find the area under a curve (which will soon be useless since definite integration is so much easier), why do I need yet another, more complex way to find it? Am I missing out on a drastic concept here that I need to know, or are these just kinda extra?

Shreyas Shridharan
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    What you say is true for continuous functions, but alas, not all functions are continuous. We need the general definition so that the integral will behave properly in the general case. That is, we have to restrict which functions are integrable if we want the integral to have useful properties. – saulspatz Oct 26 '19 at 18:17
  • So what is the definition of the integral then? The upper, lower, or midpoint Riemann sum? – Shreyas Shridharan Oct 26 '19 at 18:18
  • The integral converges if the limit of the upper sums is the same as the limit of the lower sums, so when the integral exists, you can use either one. Same goes for the midpoint of the interval. – saulspatz Oct 26 '19 at 18:21
  • Alright, but what I'm asking is why we need to know the midpoint or lower Reimann sum formula since it equals the simpler upper Riemann sum formula when n approaches infinity. Is this needed elsewhere in calculus? – Shreyas Shridharan Oct 26 '19 at 18:23
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    No, it's just used for developing the theory. You'll learn a better way to compute integrals, if you haven't already. – saulspatz Oct 26 '19 at 18:32
  • Oh, okay. Thanks for answering! – Shreyas Shridharan Oct 27 '19 at 00:55

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If the function is integrable then it really does not matter which points you pick as long as the $\delta x $ goes to zero as $n$ goes to infinity.

Not every function is integrable so one way to prove that a given function is not integrable is to show that there are two Riemann sums which tend to different limits.

That is why we learn different methods.

Mohammad Riazi-Kermani
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