# So many Integrals – I

Standard

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 $f$ be a positive continuous function defined on an interval $[a, b],\quad a, b$ being real numbers. Let $P : a = x_0 < x_1 < x_2<\ldots < x_n = b$, $n$ being an integer, be a partition of the interval $[a, b]$ and form the sum

$S_p = \sum_{i=1}^n (x_i - x_{i-1}) f(t_i)$

where $t_i \in [x_{i-1} , x_i]f$ be such that $f(t_i) = \text{Minimum} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$

By adding more points to the partition $P$, we can get a new partition, say $P'$, which we call a ‘refinement’ of $P$ and then form the sum $S_{P'}$.  It is trivial to see that $S_P \leq S_{P'} \leq \text{Area bounded between x-axis and function}f$

Since, $f$ is continuous (and positive), then $S_P$ becomes closer and closer to a unique real number, say $kf$, as we take more and more refined partitions in such a way that $|P| := \text{Maximum} \{x_i - x_{i-1}, 1 \leq i \leq n\}$ becomes closer to zero. Such a limit will be independent of the partitions. The number $k$ is the area bounded by function and x-axis and we call it the Cauchy integral of $f$ over $a$  to $b$. Symbolically, $\int_{a}^{b} f(x) dx$ (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 $f$ to prove that the sum $S_p$ indeed converges to a unique real number.

In 1851, a German mathematician, Georg Friedrich Bernhard Riemann gave a more general definition of integral.

Let $[a,b]$ be a closed interval in $\mathbb{R}$. A finite, ordered set of points $P :\{ a = x_0 < x_1 < x_2<\ldots < x_n = b\}$, $n$ being an integer, be a partition of the interval $[a, b]$. Let, $I_j$ denote the interval $[x_{j-1}, x_j], j= 1,2,3,\ldots , n$. The symbol $\Delta_j$ denotes the length of $I_j$. The mesh of $P$, denoted by $m(P)$, is defined to be $max\Delta_j$.

Now, let $f$ be a function defined on interval $[a,b]$. If, for each $j$, $s_j$ is an element of $I_j$, then we define:

$S_P = \sum_{j=1}^n f(s_j) \Delta_j$

Further, we say that $S_P$ tend to a limit $k$ as $m(P)$ tends to 0 if, for any $\epsilon > 0$, there is a $\delta >0$ such that, if $P$ is any partition of $[a,b]$ with $m(P) < \delta$, then $|S_P - k| < \epsilon$ for every choice of $s_j \in I_j$.

Now, if $S_P$ tends to a finite limit as $m(P)$ tends to zero, the value of the limit is called Riemann integral of $f$ over $[a,b]$ and is denoted by $\int_{a}^{b} f(x) dx$

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 $f$ be a bounded function defined on an interval $[a, b],\quad a, b$ being real numbers. Let $P : a = x_0 < x_1 < x_2<\ldots < x_n = b$, $n$ being an integer, be a partition of the interval $[a, b]$ and form the sum

$S_P = \sum_{i=1}^n (x_i - x_{i-1}) f(t_i), \quad \overline{S}_P =\sum_{i=1}^n (x_i - x_{i-1}) f(s_i)$

where $t_i,s_i \in [x_{i-1} , x_i]$ be such that

$f(t_i) = \text{sup} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$,

$f(s_i) = \text{inf} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$

The sums $S_P$ and $\overline{S}_P$ represent the areas and  $S_P \leq \text{Area bounded by curve} \leq \overline{S}_P$. Moreover, if $P'$ is a refinement of $P$, then

$S_p \leq S_{P'} \leq \text{Area bounded by curve} \leq \overline{S}_{P'} \leq \overline{S}_{P}$

Using the boundedness of $f$, one can show that $S_P, \overline{S}_P$ converge as the partition get’s finer and finer, that is $|P| := \text{Maximum}\{x_i - x_{i-1}, 1 \leq i \leq n\} \rightarrow 0$, to some real numbers, say $k_1, k_2$ respectively. Then:

$k_l \leq \text{Area bounnded by the curve} \leq k_2$

If $k_l = k_2$ , then we have $\int_{a}^{b} f(x) dx = k_l = k_2$.

There are two more flavours of integrals which I will discuss in next post. (namely, Stieltjes Integral and Lebesgue Integral)