For a bounded function on a (finite) interval
and a subdivision of one can define two special
stepfunctions: and (upper and lower stepfunction) as follows.
Let
. Then for all
and
satisfying
one puts
and
respectively.
EXAMPLE 2.1. (cf. EXAMPLE 1.4.)
![$\scriptstyle>$](img94.gif) f:=x->x*sin(x)^2: n:=10: S:=[seq(1.+2*`i`/n,`i`=0..n)]:
![$\scriptstyle>$](img95.gif) integral_plot(f,S,'riemann');
5
By definition the graph of lies below that of
and above the one of . Hence the area below the graph of
must satisfy
Here lower (resp. upper) is the area
under the graph of (resp. ). In our example it means the values
![$\scriptstyle>$](img105.gif) lower(f,S);
![$\scriptstyle>$](img107.gif) upper(f,S);
If we make the subdivision finer and finer the approximation of by upper and lower should be better and better.
![$\scriptstyle>$](img110.gif) n:=25: S:=[seq(1.+2*`i`/n,`i`=0..n)]:
![$\scriptstyle>$](img111.gif) integral_plot(f,S,'riemann');
![$\scriptstyle>$](img113.gif) lower(f,S);
![$\scriptstyle>$](img115.gif) upper(f,S);
Let us denote the subdivision devided into subintervals by . Now we must realize that ``the area under the graph of ''
has never been properly defined (though it is a
very intuitive notion). The notions introduced here, however, give a
good opportunity.
- Definition:
is called (Riemann-)
integrable if there exists a real number such that
The number is called the ``integral of over ''.
- Notations:
-
commonly or sometimes in text form
int .
Subsections
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