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 $f$ 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
$\int_a^b f(x)dx = F(b) – F(a).$
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.
Best 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://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.