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

$\displaystyle \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)$.

$\displaystyle \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.

$\displaystyle F'(x)$ $\displaystyle = \lim_{h\to 0} \frac{F(x+h) - F(x)}{h}$    
  $\displaystyle = \lim_{h\to 0} \left(\int_{a}^{x+h} f(t)dt - \int_{a}^{x} f(t)dt\right)/h$    
  $\displaystyle = \lim_{h\to 0} \left(\int_{x}^{x+h} f(t)dt\right)/h$    

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

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

which proves the theorem. $ \qedsymbol$

William Stein 2006-03-15