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1) Fundamental Theorem Of Calculus II $$ \int_{a}^{b}f'(x) = f(b) - f(a)$$ 2) Net Change Theorem $$ \int_{a}^{b}f'(x) = f(b) - f(a)$$

They are the same, why have two?

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    $\begingroup$ I never heard of the 'net change theorem', do you have a reference? $\endgroup$ – copper.hat Apr 2 '16 at 4:55
  • $\begingroup$ @copper.hat i.imgur.com/frRiCNs.png $\endgroup$ – AlanSTACK Apr 2 '16 at 4:57
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    $\begingroup$ @copper.hat This theorem only exists in calculus textbooks, since it is basically a duplicate of FToC. $\endgroup$ – Henricus V. Apr 2 '16 at 5:01
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They are the same, but "net change theorem" is arguably a better/more descriptive name.

(I like this name because it emphasizes the intuition that we are adding up a bunch of tiny or "infinitesimal" changes to obtain the net change. I think many calculus classes fail to convey this intuition.)

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    $\begingroup$ no, but "(the two) fundamental theorem of derivative and integrals" would probably be a better name than "fundamental theorem of calculus" $\endgroup$ – reuns Apr 2 '16 at 6:37
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Context matters. Mathematically they are the same but people may use them when referring to differing things. For example the net change theorem may be better written as: $$\int_a^br(t)dt=Q(b)-Q(a)$$ When discussing it like this r(t) is specifically the rate of flow for some "charge" Q. And the net charge is $\Delta Q= Q(b)- Q(a)$

A similar more physical example of this is the concept of voltage and electro-motor force. Both are the same thing but different groups solving different problems came to the same conclusion more or less independently and as a result we have two conventions that have not unified.

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