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Despite there're some related topics to this one, I'm still struggling to figure out how mathematicians came to use antiderivatives ( F(b) - F(a) in a definite integral formula) in order to find out the area under the curve?

I know how to calculate both derivative and antiderivative, but this knowledge comes from learning the rigorous formulas by heart; but what I would like to understand is the train thought that has lead to realization that we need to use antiderivatives while integrating.

For instance, I look at the simple function graph y=2x. I know the algorithm of steps I need to take to find the area, say, between 0 and 2. However, why antiderivative? Why do we work not with the f(0) and f(2), but first find the "parent" function and then work with F(0) and F(2) ?

What kind of reasoning should I follow to bind such things like "infinite number of rectangles" and antiderivative of the function at the given values?

I'm studying Algebra and Calculus on my own, so that's the reason for, perhaps, a stupid question. Still, thanks in advance for any help.

mohican93
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  • This is the fundamental theorem of calculus. It states that the values of the indefinite integral can be used to calculate the definite integral (which measures the area under the curve). – Michael Burr Jan 05 '18 at 20:15
  • Yes, I read the FTOC, but it still isn't obvious to me, why do we integrate, or find antiderivative for that? Why, e.g., not differentiating it further and further, but coming exactly to this solution? When we look at some function curve, what is pointing out to taking the antiderivative? – mohican93 Jan 05 '18 at 20:18
  • Related: https://math.stackexchange.com/questions/2226732/proof-that-the-area-under-a-curve-is-the-definite-integral-without-the-fundamen/2226894#2226894 , https://math.stackexchange.com/questions/1991575/why-cant-the-second-fundamental-theorem-of-calculus-be-proved-in-just-two-lines/1991585#1991585 – Ethan Bolker Jan 05 '18 at 20:22
  • The Wikipedia page has some intuitive explanations https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Physical_intuition – Air Conditioner Jan 05 '18 at 20:28
  • Because the curvy area can very nicely be approximated with thin rectangular stripes ... which make the definition of the Riemann integral – Hagen von Eitzen Jan 05 '18 at 20:34
  • Convince yourself by simple examples, that if you graph a (positive) velocity function $v(t)$ of a particle moving on a line then the area underneath it from $t = 0$ to$ t = 5$ is the distance traveled by the particle for $0<t<5$. Start with a constant velocity function, then try a linear one, and for more general functions, break it into thin rectangular strips and think about adding up the distance traveled in all the small time intervals. – Ned Jan 05 '18 at 21:16
  • Please note this: the only answer to a question “why is $A$ equal to $B$?” is the proof. There are transparent proofs, and there are opaque ones, especially in this matter. Clearly you were exposed to one of the opaque proofs, so you should look for a transparent one. – Lubin Jan 05 '18 at 21:59
  • "Why do we work not with the f(0) and f(2), but first find the "parent" function and then work with F(0) and F(2) ?" Nobody forbids you to work with $f(0)$ and $f(2)$, go ahead! –  Jan 05 '18 at 22:00
  • @Professor Vector the definition does. For some function, f(1) is 2, whereas F(1) is 4. We get different answers, so I don't see how I can substitute F for f. – mohican93 Jan 05 '18 at 22:06
  • And that's not reason enough for you to work with the function giving the correct result? –  Jan 05 '18 at 22:08
  • I've asked the question about how one logically explores the new function. I'm not working in the way "wow coincidence, some another function (F) is giving the correct result, so I'm going to work with this one". – mohican93 Jan 05 '18 at 22:21

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