# how could we put the limits in definite integration

24 Points
13 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 :

1. (n times) , where is a constant
2. , where is a constant

Be sure to ask if anything's not clear.

Regards and Best of Luck,

Rajat