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 .
Preservation of Inequality. If f and g are integrable on and for every x in , then
One Final Important Property: