]>
Home | 18.013A | Chapter 20 |
||
|
Suppose we have a non-negative function of the variable , defined in some domain that includes the interval with .
If is sufficiently well behaved, there is a well defined area enclosed between the lines and the curve .
That area is called the definite integral of between and (of course only for those functions for which it makes sense).
It is usually written as
If lies between and we obviously have
In order to make this equation hold for arbitrary , we require that when is less than the symbols above represent the negative of the area indicated.
Where the function is sometimes negative, we define the definite integral and the same symbols to represent the area between the x-axis and where is positive minus the area between the two when is negative (when is less than ).
To make this definition mathematical we must give a procedure for computing the area, at least in theory, and some indication of what functions we can and cannot define it for.
Here is called the integrand , and it is said to be integrated " ".
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 is a constant , then the area in question will be a rectangle and the area will be .
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.
|