The definite integral

In our discussions about area and averages , we found similar expressions. Namely, if $ f(x) $ was a function defined over an interval $ [a,b] $ , we would break the interval into n pieces of width $ \Delta x = \frac{b-a}{n} $ and call the points $ x_i = a + i \Delta x $ . Then we were led to consider

\[ \lim_{n\to\infty} \sum_{i=0}^{n-1} f(x_i) \Delta x \]

In fact, we will frequently see this type of thing and so we would like to devise a shorthand for it. We will call this expression the definite integral of $ f(x) $ over the inteval $ [a,b] $ and write it as

\[ \int_a^b f(x) ~dx = \lim_{n\to\infty} \sum_{i=0}^{n-1} f(x_i) \Delta x \]

This notation is designed to be suggestive: the integral sign $ \int $ is meant to remind you of an exaggerated $ \Sigma $ . That means, first and foremost, that integrals are very much like sums. What follows the integral sign-- $ f(x) dx $ --should also remind you of $ f(x_i)\Delta x $ .

We have already seen that the area under the graph of a positive function is given by

\[ A = \int_a^b f(x)~dx \]

and that the average value of a function over an interval is given by

\[ \bar{f} = \frac 1{b-a}\int_a^b f(x)~dx \]


Properties of the definite integral

Since definite integrals are formed from a process of summing, we might expect that they behave very much like sums. Here are some of the more useful properties of definite integrals.
  1. Linearity: If c and d are constants, then

    \[ \int_a^b (cf(x) + dg(x))~dx = c\int_a^b f(x)~dx + d\int_a^b g(x) ~dx \]
    This is merely a reflection of the fact that sums behave in this fashion due to the distributive nature of addition.

  2. If we have values $ a < b < c $ , then

    \[ \int_a^c f(x)~dx = \int_a^b f(x) ~dx + \int_b^c f(x)~dx \]

    The reason for this property is easily seen if you interpret the definite integrals as areas under the graph of $ f(x) $ .

Now that we have seen the definite integral in a few different places, we would like to have a means available to us to compute it. In fact, we have done this for a few nice functions when we computed the area under the graph of a function. However, this was a pretty complicated and involved process just for these simple functions. It is a bit like when we were learning to differentiate functions: we had a formal procedure involving a limit but we quickly devised some shortcuts to help us.

We will next find a shortcut for computing definite integrals. However, rather than being a process which merely simplifies our computations, our shortcut will be an important theorem which expresses a deep, underlying relationship between integration and differentiation. In fact, its title---The Fundamental Theorem of Calculus---indicates its importance.