Submitted to HEJ Manuscript no.: ANM-010201-A

The Riemann integral

For a bounded function $f$ on a (finite) interval $[a,b]$ and a subdivision $S$ of $[a,b]$ one can define two special stepfunctions: $U$ and $L$ (upper and lower stepfunction) as follows.

Let $S:=[a_{0},\ldots,a_{n}]$. Then for all $i\ (1\leq i\leq n)$ and $x$ satisfying $a_{i - 1}<x\leq a_{i}$ one puts

$$U(x) = \mbox{sup}(f(y): a_{i - 1}<y\leq a_{i})$$

and

$$L(x) = \mbox{inf}(f(y): a_{i - 1}<y\leq a_{i}),$$

respectively.

EXAMPLE 2.1. (cf. EXAMPLE 1.4.)

$>$ f:=x->x*sin(x)^2: n:=10: S:=[seq(1.+2*`i`/n,`i`=0..n)]:

$>$ integral_plot(f,S,'riemann');

 

Figure 5:

5

By definition the graph of $f$ lies below that of $U$ and above the one of $L$. Hence the area $I$ below the graph of $f$ must satisfy

$$\mbox{\bf lower}(f,S)\leq I\leq \mbox{\bf upper}(f,S).$$

Here lower (resp. upper) is the area under the graph of $L$ (resp. $U$). In our example it means the values

$>$ lower(f,S);

$$1.992058554$$

$>$ upper(f,S);

$$2.521327986$$

If we make the subdivision finer and finer the approximation of $I$ by upper and lower should be better and better.

$>$ n:=25: S:=[seq(1.+2*`i`/n,`i`=0..n)]:

$>$ integral_plot(f,S,'riemann');

 

Figure 6:

$>$ lower(f,S);

$$2.157692545$$

$>$ upper(f,S);

$$2.369400318$$

Let us denote the subdivision $S$ devided into $n$ subintervals by $S_n$. Now we must realize that ``the area under the graph of $f$'' has never been properly defined (though it is a very intuitive notion). The notions introduced here, however, give a good opportunity.

Definition:

$f$ is called (Riemann-) integrable if there exists a real number $I$ such that

$$I=\lim_{n\rightarrow \infty} {\bf lower}(f,S_n) = \lim_{n\rightarrow \infty} {\bf upper}(f,S_n).$$

The number $I$ is called the ``integral of $f$ over $[a,b]$''.

Notations:

$\int_{a}^{b}f(x)\,dx$ commonly or sometimes in text form
int$(f(x),x=a..b)$.

Submitted to HEJ Manuscript no.: ANM-010201-A