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        how could we put the limits in definite integration

10 years ago

24 Points
							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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• 51 Video Lectures
• Revision Notes
• Test paper with Video Solution
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Previous Year Exam Questions