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.
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”).
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
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)