The Fundamental Theorem of Calculus

Let $f$ be a continuous function on the interval $[a,b]$. The following theorem is incredibly useful in mathematics, physics, biology, etc.

Theorem 2.1.3   If $F(x)$ is any differentiable function on $[a,b]$ such that $F'(x) = f(x)$, then

$$\int_{a}^{b} f(x) dx = F(b) - F(a).$$

One reason this is amazing, is because it says that the area under the entire curve is completely determined by the values of a (``magic'') auxiliary function at only $2$ points. It's hard to believe. It reduces computing (2.1.2) to finding a single function $F$, which one can often do algebraically, in practice.Whether or not one should use this theorem to evaluate an integral depends a lot on the application at hand, of course. One can also use a partial limit via a computer for certain applications (numerical integration).

Example 2.1.4   I've always wondered exactly what the area is under a ``hump'' of the graph of $\sin$. Let's figure it out, using $F(x) = -\cos(x)$.

$$\int_{0}^\pi \sin(x) dx = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 2.$$

But does such an $F$ always exist? The surprising answer is ``yes''.

Theorem 2.1.5   Let $F(x) = \int_{a}^{t} f(t) dt$. Then $F'(x) = f(x)$ for all $x\in [a,b]$.

Note that a ``nice formula'' for $F$ can be hard to find or even provably non-existent.

The proof of Theorem 2.1.5 is somewhat complicated but is given in complete detail in Stewart's book, and you should definitely read and understand it.

Proof. [Sketch of Proof] We use the definition of derivative.

$$F'(x) = \lim_{h\to 0} \frac{F(x+h) - F(x)}{h}$$

$$= \lim_{h\to 0} \left(\int_{a}^{x+h} f(t)dt - \int_{a}^{x} f(t)dt\right)/h$$

$$= \lim_{h\to 0} \left(\int_{x}^{x+h} f(t)dt\right)/h$$

Intuitively, for $h$ sufficiently small $f$ is essentially constant, so $\int_{x}^{x+h} f(t)dt \sim hf(x)$ (this can be made precise using the extreme value theorem). Thus

$$\lim_{h\to 0} \left(\int_{x}^{x+h} f(t)dt\right)/h = f(x),$$

which proves the theorem. $\qedsymbol$