Definition of a Definite Integral

 

 

Let f be defined on the closed interval  and divide this interval into n subintervals of equal width . Let  denote the endpoints of these subintervals and choose sample points  in these subintervals so that  lies in the ith subinterval . Then the definite integral of f from a to b (if it exists) is

 

 

 

Note:  The definite integral is a number; it does not depend on x.

 

Note: If f is continuous it can be shown that the above limit exists no matter how the sample points are chosen. It is convenient to take the sample points to be the right endpoints . Then the definition of an integral becomes

 

 

 

Note:  The sum

 

 

 

is called a Riemann sum. If f happens to be positive, this sum can be interpreted as a sum of areas of approximating rectangles. In fact, in this case (with f continuous on  ), the definite integral  can be interpreted as the area under the curve  from a to b.

            If f takes on positive and negative values on , then the definite integral can be interpreted as a net area, the difference between the area above the x-axis  and the area below :

 

 

Note:  The interval [a, b] need not be divided into subintervals of equal length in order to define the definite integral; using subintervals of equal length just makes things easier.

 

 

As examples, evaluate the following definite integrals.

 

 

For the second example above, first evaluate a Riemann sum taking the sample points to be right endpoints and , , and .

 

 

Two Special Definite Integrals

 

 

If f is defined at , then .

 

If f is integrable on , then .

 

 

Some Properties of Definite Integrals

 

 

If f and g are integrable on  and c is a constant, then the functions of cf and  are integrable on , and

 

 

Preservation of Inequality. If f and g are integrable on  and  for every x in , then

 

 

 

One Final Important Property: