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How can I interpret indefinite integrals in term of area? I'm looking at the second fundamental theorem of calculus:

$$\int_a^b f(x)\mathrm dx = F(b) - F(a)$$

What is the meaning of F(b)? Does F(b) mean the area from -infinity to b? If that's the case I can see how $F(b) - F(a)$ is the area from a to b, but it's not clear to me how the indefinite integral evaluated at b can be interpreted as that.

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  • $\begingroup$ The function $F$ is any function such that $F'(x)=f(x)$, $x\in (a,b)$. $\endgroup$ Commented Oct 19, 2022 at 21:54
  • $\begingroup$ $F(q)$ means the area from ANY PARTICULAR point to $q$. It doesn't matter which point, as long as it is the same for all usages. That's what allows $F(b) - F(a)$ to work. $\endgroup$ Commented Oct 19, 2022 at 22:33

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In your equation $F$ is not an indefinite integral. It is some particular one of the functions whose derivative is $f$. Formally speaking, the "indefinite integral" is the set of those functions. There is no such thing as $F(b)$ for an indefinite integral.

Any two of those functions differ by a constant. Geometrically, you can think of their graphs as vertical translations of any one of them.

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  • $\begingroup$ I'm asking more how to interpret based on area, but I think johnnyb's answer makes sense. Also, every source I see says that "F" is an indefinite integral $\endgroup$ Commented Oct 21, 2022 at 19:43

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