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 ().
- Definition:
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
.
- Definition:
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
.
Example:
For positive integer “”,
is a partition of
is a partition of
is not a partition of
as
.
- Definition:
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
.
i,e., .
- Definition:
Upper Sum: Let, be a bounded function on
and
be a partition over
.
Then “” is bounded on every
.
Let,
Then, defined as upper sum or Darboux upper sum. r
Lower sum: Let, be a bounded function on
and
be a partition over
.
Let,
Then,
defined as lower sum or Darboux lower sum.
r
Notation:
Upper Sum:
For the function “” over the partition”
” on
,
Then,
Lower Sum: For the function “” over the partition “
” on
,
Then,
Facts:
- If
and
are equal when
is a constant function.
2.
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
.
- Definition
Refinement of a partition:
Let, and
be two partitions of
such that if
then
is called refinement of the partition “
”.
- Definition:
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 “
”
Then
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
Particular fact:
For any real valued bounded function on and any two partitions
,
of
Definition:
Riemann Integrability:
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
Example:
Let
Check
Solution: Let
be a partition of where “
” is a positive integer.
Then “” divides
into “
” sub-intervals
Here is monotonically increasing and continuous function on
Also
So,
And
Here,
Hence,
And
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
.
Then
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
Also
Therefore
Hence
Conversely (Sufficient condition)
Suppose for any positive number there exists a partition
on
such that
Since,
and
Also
Therefore
Hence
Therefore
Hence is Riemann Integrable on
Hence proved.
Remark:-
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
s
Here contains rational as well as irrational points for each
So,
Then,
Also
Then
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.
(Why?)
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
.
But
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.
Let’s start
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
Hence
Therefore we can do this now
Therefore,
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
.
Then
- If
and
is integrable on
and
for all values of
in
.
Then
- If
is integrable on
Then
Problems related with these theories:-
Example.
Show that
To prove this we are going to use all of the theories which we learnt till now,
Let
Here for all
For all
For all
Therefore let for all
and
Here
for all
So therefore
Now
`
And
Therefore
Hence proved.
Now some work sheet problems:-
- Let
Check is Riemann Integrable or not.
- Try to prove these problems
i)
- ii)
iii)
- Try to show that
Bibliography
- 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.
Thank you………