Why the fundamental theorems of calculus are fundamental?

Question

I can tell why the fundamental theorem of arithmetic and the fundamental theorem of algebra are fundamental, but, indeed, I cannot convince myself why the fundamental theorems of calculus are fundamental.

Any feedback is highly appreciated!

Answer 1

I agree with Lolman: sometimes we believe that a theorem is fundamental only because we call it so for historical reasons. In Italy we call it "The Torricelli-Barrow Theorem", probably because we don't think it is so... fundamental!

Apart from jokes, this theorem is important because it establishes a link between to concepts that have little in common: the concept of antiderivative and the concept of (Riemann) integral. Basically, we tend to believe that antiderivatives are (indefinite) integrals only because the fundamental theorem of calculus holds. Just pick one of the few textbooks that write $D^{-1}f$ instead of $\int f$, and you'll realize that definite and indefinite integrals are rather distinct subjects.

But then comes the FToC, and we learn that (for continuous functions) $x \mapsto \int_a^x f$ is an antiderivative of $f$, and, as a consequence, that $\int_a^b f = F(b)-F(a)$ for any antiderivative $F$ of $f$. The FToC should be called The Very Important Theorem of Calculus nowadays, since it has been generalized by measure theory. However, mathematical analysis wouldn't go very far without it.

Answer 2

The important reason is that it connects differentiation and integration, hence, as a corollary, difficult to calculate definite integral with much easier to calculate indefinite integral.