Page 245 - DMTH401_REAL ANALYSIS

Unit 20: The Riemann Integration

The second view is related to the limiting process. It gives rise to an integral which is the limit Notes
of all the values of a function in an interval. This is the integral of a function f(x) over an interval
[a,b,]. It is called the definite integral and is denoted by

b
 f(x)dx
a
The definite integral is a number since geometrically it corresponds an area of a region enclosed
by the graph of a function.
Although both the notions of integration are closely related, yet, you will see later, the definite
integral turns out to be a mare fundamental concept. In fact, it is the starting point for some
important generalizations like the double integrals, triple integrals, line integrals etc., which
you may study on Advanced Calculus.
The integral in the anti-derivative sense was given by Neyrtan. This notion was found to be
adequate so long as the functions to be integrated were continuous. But in the early 19th century,
Fourier brought to light the need for making integration meaningful for the functions that are
not continuous. He came across such functions in applied problems. Cauchy formulated rigorous
definition of the integral of a function. He essentially provided a general theory of integration
but only for continuous functions. Cauchy's theory of Integration for continuous functions is
sufficient for piece-wise continuous functions as well as for the functions having isolated
discontinuities. However, it was G.B.F. Riemann [1826-1866] a German mathematician who
extended Cauchy's integral to the discontinuous functions also. Riernann answered the question
"what is the meaning of  f(x) dx?"

The concept of definite integral was given by Riemann in the middle of the nineteenth century.
That is why, it is called Riemann Integral. Towards the end of 19th Century, T.J. Stieltjes [1856-
1894] of Holland, introduced a broader concept of integration replacing certain linear functions
used in Riemann Integral by functions of more general forms. In the beginning of this century,
the notion of the measure of a set of real numbers paved the way to the foundation of modern
theory of Lebesgue Integral by an eminent French Mathematician H. Lebesgue [1875-1941], a
beautiful generalisation of Riemann Integral which you may study in some advanced courses of
Mathematics. In this unit, the Riemann Integral will be defined without bringing in the idea of
differentiation. As you have been go through the usual connection between the Integration and
Differentiation. Just by applying the definition, it is not always easy to test the integrability of
a function. Therefore, condition of integrability will be derived with the help of which it becomes
easier to discuss the integrability of functions. Then just as in the case of continuity and
derivability, we will also consider algebra of integrable functions. Finally, in this unit, second
definition of integral as the limit of a sum will be given to you and you will be shown the
equivalence of the two definitions.

20.1 Riemann Integration

The study of the integral began with the geometrical consideration of calculating areas of plane
figures. You know that the well-known formula for computing the area of a rectangle is equal to
the product of the length and breadth of the rectangle. The question that arises from this formula
is that of finding the correct modification of this formula which we can apply to other plane
figures. To do so, consider a function defined on a closed interval [a,b] of the real line, which
assumes a constant value K  0 throughout the interval. The graph of such a function gives rise
to a rectangular region bounded by the X-axis and the ordinates x = a, x = b as shown in the
Figure 20.1.

LOVELY PROFESSIONAL UNIVERSITY 239