
We all know that, area is the basis of integration theory, just as counting is basis of the real number system. So, we can say:
An integral is a mathematical operator that can be interpreted as an area under curve.
But, in mathematics we have various flavors of integrals named after their discoverers. Since the topic is a bit long, I have divided it into two posts. In this and next post I will write their general form and then will briefly discuss them.
Cauchy Integral

Newton, Leibniz and Cauchy (left to right)
This was rigorous formulation of Newton’s & Leibniz’s idea of integration, in 1826 by French mathematician, Baron Augustin-Louis Cauchy.
Let be a positive continuous function defined on an interval
being real numbers. Let
,
being an integer, be a partition of the interval
and form the sum
where be such that
By adding more points to the partition , we can get a new partition, say
, which we call a ‘refinement’ of
and then form the sum
. It is trivial to see that
Since, is continuous (and positive), then
becomes closer and closer to a unique real number, say
, as we take more and more refined partitions in such a way that
becomes closer to zero. Such a limit will be independent of the partitions. The number
is the area bounded by function and x-axis and we call it the Cauchy integral of
over
to
. Symbolically,
(read as “integral of f(x)dx from a to b”).
Riemann Integral

Riemann
Cauchy’s definition of integral can readily be extended to a bounded function with finitely many discontinuities. Thus, Cauchy integral does not require either the assumption of continuity or any analytical expression of to prove that the sum
indeed converges to a unique real number.
In 1851, a German mathematician, Georg Friedrich Bernhard Riemann gave a more general definition of integral.
Let be a closed interval in
. A finite, ordered set of points
,
being an integer, be a partition of the interval
. Let,
denote the interval
. The symbol
denotes the length of
. The mesh of
, denoted by
, is defined to be
.
Now, let be a function defined on interval
. If, for each
,
is an element of
, then we define:
Further, we say that tend to a limit
as
tends to 0 if, for any
, there is a
such that, if
is any partition of
with
, then
for every choice of
.
Now, if tends to a finite limit as
tends to zero, the value of the limit is called Riemann integral of
over
and is denoted by
Darboux Integral

Darboux
In 1875, a French mathematician, Jean Gaston Darboux gave his way of looking at the Riemann integral, defining upper and lower sums and defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller.
Let be a bounded function defined on an interval
being real numbers. Let
,
being an integer, be a partition of the interval
and form the sum
where be such that
,
The sums and
represent the areas and
. Moreover, if
is a refinement of
, then
Using the boundedness of , one can show that
converge as the partition get’s finer and finer, that is
, to some real numbers, say
respectively. Then:
If , then we have
.
There are two more flavours of integrals which I will discuss in next post. (namely, Stieltjes Integral and Lebesgue Integral)