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So, obviously it gives the area between the function and the x-axis. What I'm wondering is, are there any analogs to the way we can find things with derivatives? (i.e. extrema, intervals of increasing/decreasing, concavity, inflection points, etc.) Or is there something I'm missing about what the area can tell us?

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  • $\begingroup$ If the integral is increasing, the function is positive; just like if the derivative is positive, the function is increasing. All of those theorems are pretty much reversible. $\endgroup$ – G Tony Jacobs Apr 11 '18 at 1:11
  • $\begingroup$ @GTonyJacobs Yeah, but... to find if the integral is increasing, you'd just take the derivative of that, and that's just the function... I mean your answer is perfectly valid but I just feel like I want there to be more than just going back to the original function. Like, to do things with derivative like that, you don't integrate back to the function. $\endgroup$ – siren5474 Apr 11 '18 at 1:16
  • $\begingroup$ Taking a derivative isn't the only way to show that something is increasing. Some functions are integrated empirically, and you don't have a formula, even. Nevertheless, I understand your question. In the textbook, there's a whole section or three on how we can get information about a function from its derivative(s). Why not a similar part of the curriculum for integrals? $\endgroup$ – G Tony Jacobs Apr 11 '18 at 1:42
  • $\begingroup$ @GTonyJacobs Yeah, that's exactly what I'm asking. Surely something like that stuff exists and is just as practical, right? $\endgroup$ – siren5474 Apr 11 '18 at 1:46

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