# How is the Net Change Theorem different from Fundamental Theorem Of Calculus II

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?

• I never heard of the 'net change theorem', do you have a reference? – copper.hat Apr 2 '16 at 4:55
• @copper.hat i.imgur.com/frRiCNs.png – AlanSTACK Apr 2 '16 at 4:57
• @copper.hat This theorem only exists in calculus textbooks, since it is basically a duplicate of FToC. – Henricus V. Apr 2 '16 at 5:01

## 2 Answers

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.)

• no, but "(the two) fundamental theorem of derivative and integrals" would probably be a better name than "fundamental theorem of calculus" – reuns Apr 2 '16 at 6:37

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.