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.