]> 20.1 Area and Notation

## 20.1 Area and Notation

Suppose we have a non-negative function $f$ of the variable $x$ , defined in some domain that includes the interval $[ a , b ]$ with $a < b$ .

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 $f d x$ between $x = a$ and $x = b$ (of course only for those functions for which it makes sense).

It is usually written as

$∫ x = a x = b f ( x ) d x$

If $c$ lies between $a$ and $b$ we obviously have

$∫ x = a x = c f ( x ) d x + ∫ x = c x = b f ( x ) d x = ∫ x = a x = b f ( x ) d x$

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 x-axis 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 " $d s$ ".

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 $c ( b − a )$ .

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.