Fundamental Theorem of Calculus

So far, $\int f(x)\,dx$ has meant the INDEFINITE integral -- a family of antiderivatives. A

definite integral $\int_a^b f(x)\,dx$ is a different object: a specific NUMBER, informally

the (signed) area between the curve $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$. The

Fundamental Theorem of Calculus is the remarkable fact connecting the two -- that this area

can be computed using nothing more than an antiderivative found by the very techniques of

Sections 2-6, without ever summing areas directly. (As per the syllabus, this theorem is stated

here without proof.)

The area function. For a continuous function $f$ on $[a,b]$, define

the (signed) area swept out from $a$ up to the variable point $x$. $A(x)$ is itself a function of

its upper limit $x$.

First Fundamental Theorem of Calculus. $A(x)$ is differentiable, and $A'(x)=f(x)$ -- that is,

the area function is ITSELF an antiderivative of $f$. In words: differentiating the area swept

out so far, with respect to the upper limit, gives back the original function -- confirming, at

the level of areas, that integration and differentiation are exact inverses of each other

(Section 1).

Second Fundamental Theorem of Calculus. If $F$ is ANY antiderivative of $f$ (i.e.

$F'=f$, found by ordinary indefinite integration), then

often written $\big[F(x)\big]_a^b$. This is the working form used throughout Sections 8-9: since

$A(x)$ and $F(x)$ are both antiderivatives of $f$, they differ only by a constant (Section 1), and

that constant cancels out in the subtraction $F(b)-F(a)$ -- so it makes no difference WHICH

antiderivative is used, or whether the arbitrary constant $C$ is even written down for a definite

integral.

Why this matters. The Second Fundamental Theorem is what turns every indefinite-integration …