| 释义 |
Riemann IntegralThe Riemann integral is the Integral normally encountered in Calculus texts and used by physicists andengineers. Other types of integrals exist (e.g., the Lebesgue Integral), but are unlikely to be encountered outsidethe confines of advanced mathematics texts.
The Riemann integral is based on the Jordan Measure, and defined by taking a limit of a Riemann Sum,
 | (1) |
 | (2) |
 | (3) |
 and  ,  , and  are arbitrary points in the intervals  ,  ,and  , respectively. The value  is called the Mesh Size of a partition of the interval into subintervals .
As an example of the application of the Riemann integral definition, find the Area under the curve  from 0 to . Divide  into  segments, so  , then
By induction
 | (7) |
 | (8) |
 | (9) |
 .so
See also Integral, Riemann Sum
References
Kestelman, H. ``Riemann Integration.'' Ch. 2 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 33-66, 1960. |
