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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 , ... ,
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 :
Be sure to ask if anything's not clear.
Regards and Best of Luck,
Rajat
Askiitian Expert
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