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:

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

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

If ƒ is integrable on [a, b] then

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

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 dosen't seem like such a big deal, especially when indefinite integrals can be more interesting.

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    $\begingroup$ If this doesn't excite you, can you say a bit about what does? Why does MVT or IVT excite you? You say that the first part gives a joyful union of integral and derivative, but if this was called part one and the former part two, would you be happy with this being the fundamental connection? Moreover, how do you feel about the fact that this gives a relationship between the endpoint data, and all the data in between? $\endgroup$ – BBischof Nov 2 '10 at 15:29
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    $\begingroup$ Which part is part one? $\endgroup$ – Qiaochu Yuan Nov 2 '10 at 15:38
  • $\begingroup$ MVT, IVT lets us confirm what we knew intuitively. They both really bolster ones confidence in the calculus. I suppose this theorem is the same. The difference of the areas in the space in between... but it's almost too obvious. $\endgroup$ – futurebird Nov 2 '10 at 15:44
  • $\begingroup$ As in this article, Qiaochu: en.wikipedia.org/wiki/Fundamental_theorem_of_calculus $\endgroup$ – futurebird Nov 2 '10 at 15:45

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://math.uga.edu/~pete/243integrals1.pdf (Wayback Machine)

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.

  • $\begingroup$ Thank you, this text and you comment are very helpful. $\endgroup$ – futurebird Nov 2 '10 at 15:59

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).


This is the part of the fundamental theorem that allows you to compute integrals; then you can compute areas, and with more theory even volumes, surfaces and so on. Exciting enough?

  • $\begingroup$ We establish a means of computing derivatives much earlier in most calc. courses, and both of these things are very useful but useful is not the same as "fundamental." For example, the fundamental theorem of Galois theory unites the world of fields and subfields with that of groups and subgroups. It's a bridge. The first part of the FTOC works in the same way. In the pdf Pete posted the 2nd part is more of a corr. -- and that just makes so much more sense to me. Most books present them as if they have equal weight. I don't think they do. $\endgroup$ – futurebird Nov 2 '10 at 16:10
  • $\begingroup$ @alittledon: I'm sorry that my answer wasn't that helpful. I just typed it without thinking too much, concentrating mostly on the last word of your question. $\endgroup$ – Hendrik Vogt Nov 2 '10 at 16:38
  • $\begingroup$ @a little don: it really depends on how you approach the problems. For example, one can get the 1st part as a corollary of the 2nd part! So the fact that you see the 2nd part as more of a corollary has to do with the way you approach it; they are equally likely to be seen in a way in which the 1st part is a corollary of the second. And the second part likewise establishes a connection between the two different parts/worlds of calculus: the integral calculus and the differential calculus. $\endgroup$ – Arturo Magidin Nov 2 '10 at 19:10

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