Home  18.013A  Chapter 20 


Suppose we have a nonnegative function $f$ of the variable $x$ , defined in some domain that includes the interval $[a,b]$ with $ab$ .
If $f$ is sufficiently well behaved, there is a well defined area enclosed between the lines $x=a,x=b,y=0$ and the curve $y=f(x)$ .
That area is called the definite integral of $fdx$ between $x=a$ and $x=b$ (of course only for those functions for which it makes sense).
It is usually written as
If $c$ lies between $a$ and $b$ we obviously have
In order to make this equation hold for arbitrary $c$ , we require that when $b$ is less than $a$ the symbols above represent the negative of the area indicated.
Where the function $f$ is sometimes negative, we define the definite integral and the same symbols to represent the area between the xaxis and $y=f(x)$ where $f$ is positive minus the area between the two when $f$ is negative (when $a$ is less than $b$ ).
To make this definition mathematical we must give a procedure for computing the area, at least in theory, and some indication of what functions $f$ we can and cannot define it for.
Here $f$ is called the integrand , and it is said to be integrated " $ds$ ".
Our approach to defining the area is based on the fact that we know what the area of a rectangle is, namely it is the product of the lengths of its sides. If the function $f(x)$ is a constant $c$ , then the area in question will be a rectangle and the area will be $c(ba)$ .
That's all we need to define area for a constant function.
Our task is to generalize this definition to functions that are not constant.
