Question: The 2nd part of the "Fundamental Theorem of Calculus" has never seemed as earth shaking or as **fundamental** as the first to me. Why is it "fundamental" -- I mean, the *mean value theorem*, and the *intermediate value theorems* are both pretty exciting by comparison. And after the joyful union of integration and the derivative that we find in the first part, the 2nd part just seems like a yawn. So, what am I missing?

To be clear I'm talking about this:

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Let $f$ be a real-valued function defined on a closed interval $[a,b]$ that admits an antiderivative $F$ on $[a,b]$. That is, $f$ and $F$ are functions such that for all $x$ in $[a, b]$,

$f(x) = F'(x)$

If $f$ is integrable on $[a, b]$ then

$\int\_a^b f(x)dx = F(b) - F(a).$

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I've been through the proof a few times. It makes sense to me. But, it didn't help me to see the light. To me it just looks like "OK here is how you do the definite integral." Which doesn't seem like such a big deal, especially when indefinite integrals can be more interesting.

Answer: 
It's natural that the Fundamental Theorem of Calculus has two parts, since morally it expresses the fact that differentiation and integration are mutually inverse processes, and this amounts to two statements: (i) integrating and then differentiating and (ii) differentiating and then integrating get us (essentially) back where we started.

On the other hand, many people have noticed that the two parts are not completely independent: e.g. if $f$ is continuous, then (ii) follows easily from (i). However, for discontinuous -- but Riemann integrable -- $f$, the theorem still holds, and this is what requires a nontrivial additional argument. See page 8 of

[http://alpha.math.uga.edu/~pete/243integrals1.pdf](http://alpha.math.uga.edu/%7Epete/243integrals1.pdf) ([Wayback Machine](https://web.archive.org/web/20130728121230/http://alpha.math.uga.edu/%7Epete/243integrals1.pdf))

for some discussion of this point.

I can't tell from your question how squarely this answer addresses it. If yes, and you have further concerns, please let me know.
The names "first" and "second" for the two parts of the theorem are meaningless. More correct names would be *existence* and *uniqueness*. It also is not unreasonable to separate the uniqueness statement from the formula relating definite integrals to antiderivatives, which is an algebraic *consequence* of the (analytic) uniqueness statement. The formula could be considered as a third part of the theorem, but numbering pieces of a theorem in a particular order is an uninformative nomenclature -- as is calling theorems "fundamental".

Fundamental theorem of calculus asserts existence and uniqueness of antiderivatives (solutions of the differential equation $y' = f(x)$ with given value of $y(x\_0)$ at one point). Apart from purely logical considerations there are several reasons the uniqueness theorem is important.

* Indefinite integrals of the form $\int\_p^x f(t) dt$, which are what appear in most presentations of the existence part of the theorem, in some cases do not account for all antiderivatives of $f(x)$ as the basepoint $p$ is varied over all real numbers.
* In the more precise presentation $y(x) = y(a) + \int\_a^x f(t) dt$ there is still the possibility that other processes, even more magical than integration, might be related to anti-differentiation. So it is of interest to either find these exotic species, or show that integrals give everything.
* An explicit analysis of uniqueness becomes more pressing when integrating functions with singularities, as in $\int dx/|x|^p$ for $p=1$ and $p=1/2$ (the number of integration constants changes, so this is needed for writing down solution formulas in full generality).
* the algebraic formula implied by uniqueness, $\int\_a^b f = F(b)-F(a)$, is important both as a means of computing integrals and as the basis of the notation supporting changes of integration variable (substitutions).