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        how could we put the limits in definite integration

10 years ago

							Hi,
Let me explain the concept of limits in Definite Integrals by taking a general example :

The following problems involve the limit definition of the definite integral of a continuous function of one variable on a closed, bounded interval. Begin with a continuous function on the interval .  Let ... be an arbitrary (randomly selected) partition of the interval , which divides the interval into subintervals (subdivisions).  Let ... be the sampling numbers (or sampling points) selected from the subintervals.  That is, is in , is in , is in , ... , is in , is in ,
and is in .
Define the mesh of the partition to be the length of the largest subinterval.  That is, let for and define .
The definite integral of on the interval is most generally defined to be .
For convenience of computation, a special case of the above definition uses subintervals of equal length and sampling points chosen to be the right-hand endpoints of the subintervals. Thus, each subinterval has length
equation (*) for and the right-hand endpoint formula is
equation (**) for .  The definite integral of on the interval can now be alternatively defined by .
We will need the following well-known summation rules in case od different types of functions : (n times) , where is a constant    , where is a constant Be sure to ask if anything's not clear.
Regards and Best of Luck,
Rajat

10 years ago
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