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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.

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