The deepness of the result comes from the following:
We can define two "operations" on (a certain class of) functions (namely, the functions where both definitions make sense, i.e. Riemann integrable functions):
- Given a function $f(x)$, we can find a function $F(x)$ (in fact: infinitely many, all of them differing by a constant) such that $F'(x) = f(x)$. We usually denote
$$F(x) = \int f(x) dx,$$
and given $a\le b$ two real numbers, define
$$\int_a^bf(x) dx := F(b)-F(a).$$
- Given a function $f(x)$ and two real numbers $a\le b$, we can define the Riemann integral
as a limit of certain sums on subdivisions of the interval $[a,b]$. I won't bother with the details, as I believe that you (should) know them already.
Now notice that, even though I have denoted them by the same symbols, these two definitions are different a priori. The fundamental theorem of calculus now states:
Theorem: These two definitions of
Therefore, it allows us to compute the Riemann integral of a function without having to take a limit over partitions etc., but simply (well, you get what I mean) finding an antiderivative $F(x)$ and computing the difference of its evaluation at the extremal points.