The story of integration started with I. Newton, when in the late 1660 he invented the method of inverse tangents to find areas under curves.
In 1680, G. Leibnitz discovered the process of finding tangent line to find area. Thus, they had discovered the integration, being a process of summation, was inverse to the operation of differentiation since finding tangent lines involved differences and finding areas involved summations.
In 1854, G.F Riemann formulated a new and different approach to define integral on the real line. He separated the concept from its differentiation. His approach was to examine the motivating summation and limit process of finding areas by itself.
In 1875, J.G Darboux viewed Riemann Integration in a different way. The approaches of both Riemann and Darboux demanded the integration to be bounded in its domain. It has been established that the two definitions of definite integral given by Riemann and Darboux are equivalent that is why Riemann integral are often called Darboux-Riemann Integrals.
In this article we will discuss about the integral based on Riemann and Darboux-Riemann approach will be discussed in mainly on real line ().
Partition: Let be a given closed and bounded on real line. Let be finite number of points in such that , then the finite order set
is defined to be a partition of which divides into “” closed sub-interval , for all .
Fact: Every partition “” of must include at least two points “” and “” of . In fact, is the trivial partition of .
Family of partition: Every bounded and closed interval has infinite numbers of partitions. The set or the collection of all partitions of , called family of partition of , is denoted by .Thus, if implies “” is a partition of .
For positive integer “”,
is a partition of
is a partition of
is not a partition of as .
Norm of partition:
If be a partition of and denotes the sub-intervals of (for ) having the length , for .
Then the greatest length of the sub-intervals i,e., is defined to be a norm of the partition “” and denoted by or .
Upper Sum: Let, be a bounded function on and be a partition over .
Then “” is bounded on every .
defined as upper sum or Darboux upper sum. r
Lower sum: Let, be a bounded function on and be a partition over .
defined as lower sum or Darboux lower sum.
For the function “” over the partition”” on ,
Lower Sum: For the function “” over the partition “” on ,
- If and are equal when is a constant function.
is the ‘oscillatory sum’ of “” for the partition “” on , and we write that as
- For any bounded function “” on and ,
Where, , are infimum and supremum of “” on .
Refinement of a partition:
Let, and be two partitions of such that if then is called refinement of the partition “”.
Common Refinement: If and are two partitions of then and .
So is the refinement of both and and is called common refinement of and .
- Properties of refinement of a partition:
If “” is bounded on and where “” is the refinement of “”
Definition (Darboux Lower Integral)
Let “” be a real valued function over , then is defined as lower (Darboux) integral of “” on and denoted by
Darboux Upper Integral:
For real valued function “” on , then is defined as upper (Darboux) integral of “” on and denoted by
For any real valued bounded function on and any two partitions , of
A real valued function “” on is said to be Riemann integrable on if
Then “” is Riemann integrable and are denoted that integral by
and when “” is Riemann integrable we denote that as
be a partition of where “” is a positive integer.
Then “” divides into “” sub-intervals
Here is monotonically increasing and continuous function on
So, we understand by the example how we can use the definition of Riemann Integration.
But for more complicated functions there will be difficulties to find maximum and minimum in every subinterval so, now we are going to modify the theory of “when a function is said to be a Riemann integral.”
Theorem : let be a bounded function on . A necessary and sufficient condition for inerrability of an is that for every position such that
Proof : (Necessary Part:)
Let be integrable on .
Let be any positive number. From the definition of lower and upper integral of on , we have for the given positive , there exist a partition on such that
Conversely (Sufficient condition)
Suppose for any positive number there exists a partition on such that
Hence is Riemann Integrable on
let be a bounded function on . If be a sequence if partitions of such that the sequence converges to zero and
Then is Riemann integrable on .
Now we are going to learn how to use the Remark
Example : let be a function by
Here we are going to check the integrability of
Lets try: Here is bounded on Let for we choose a partition
Here contains rational as well as irrational points for each
Hence by the Remark we can say that is not Riemann integral on .
Properties of Riemann integrable functions :
- If is monotonic then . i.e is Riemann integrable.
- If is continuous then is Riemann integrable.
- If be bounded but has finite number of discontinuous points on then
- If bounded but has infinite point on discontinuities in such that the number of limits Of these infinite discontinuous points is finite in then .
- If is integrable on , then is integrable on every closed sub interval of .
- If , and is integrable on and on then
- If is integrable on then for every then is also integrable on and further more,
- If two integrable on and
- If and then for any integrable on furthermore,
(x) If is integrable on , then on integrable. But the converse is not true.
Here, is not Riemann integrable but
is constant then it must be Riemann integrable on .
- If and are both integrable on , then also integrable on .
- If is integrable on , these is integrable on but the converse is not true (try with the same function
- If and integrable and where , then also integrable as .
- If is Riemann integrable on and where then also integrable on .
- If and bounded and closed interval. Let and integrable and continuous function such that then is integrable on .
Example 1 : – let be a bounded function on defined by
Now we are going to check is Riemann Integrable or not.
Let’s start:- Here is bounded function also is continuous on except , so is Riemann Integrable on .
As , , so is not bounded on and hence is not Riemann Integrable on and we are done.
Example 2 Consider the function such that
Here also we are going to check integrable or not in and if integrable then we will find the value of the integration.
Here the expression of is
Since for all in , is bounded on . is continuous on except the points and . So has finite number of discontinuity on then we can say that is integrable on
So hence is Riemann Integrable on
Now we are going to find the value of the integration of the function on .
Let we define
And then becomes
Therefore we can do this now
And we are done with this problem
Now try this problem with same manner such that
Try to prove is Riemann Integrable and find (Answer )
Some Inequalities related with Riemann Integration:
- If is integrable on and for all values of in .
- If and is integrable on and for all values of in .
- If is integrable on
Problems related with these theories:-
To prove this we are going to use all of the theories which we learnt till now,
Here for all
Therefore let for all
Now some work sheet problems:-
Check is Riemann Integrable or not.
- Try to prove these problems
- Try to show that
- Elementary Analysis: The theory of Calculus; Kenneth Ross.
- Improper Riemann Integration; Markos Roussos
- Modern Theories of Integration; H. Kestelman
- The Riemann Approach to Integration; Washek Pfeffer.