For a positive function, the height of each rectangle is given by a
value of f(x) for some x, and the area of the rectangle is of the form
f(x)*w where w (sometimes called "**d**elta-x") is the width of the
rectangle (a **d**ifference of x-values). Where f(x)<0 the height
of a rectangle between the graph and the x-axis would be given by -f(x),
and the term f(x)*w in a Riemann sum would give the **negative** of
the area. So in general a sum of terms of the form f(x)*w will be equal
to the total area of rectangles above the axis **minus** the total area
of those below. The **Definite Integral** of a function over an interval
is defined (if it exists) as the limit of such sums when the widths of
all the rectangles go to zero. This is equal to the area between the x-axis
and the part of the curve where f(x)>0 **minus** the area between the
x-axis and the part of the curve where f(x)<0.

There are various ways one can choose the heights of the rectangles.
In this
applet (from the IES group in Japan) we are given the choice of using
the function value at the right end, left end, or middle of each interval.
Also, in order to control the size of the error, it is a good idea to look
at the largest and smallest possible values for the rectangular areas (this
gives what are called Upper and Lower Riemann Sums), and the little "Flash"
animation that's also on our departmental home page illustrates both
of these for a particular example. The 'Analysis WebNotes' site
(at the university of Nebraska) has a section
on Riemann Sums which includes an applet in which you can choose the
intervals by hand. *IntegCalc*,
"a program to demonstrate the definition of the integral in terms of successive
approximations by Riemann sums" is one of several Java
Projects, by John Orr at the University of Nebraska.

A completely different approach to the area problem was invented in
the 17th century (mainly by Isaac Newton and Gottfried Leibnitz). The idea
was to consider the "area-so-far" function giving the area under a curve
from some fixed point to a variable endpoint (this is often called an indefinite
integral), and then to use the fact that when the right end of the
interval is moved a little bit, the change in area under the curve is approximately
equal to the height of the curve times the distance moved. So the rate
of change of area per unit distance is equal to the height. In other words
the height function is the **derivative** of the "area-so-far" function,
or conversely the "area-so-far" function is an **antiderivative**
for the height. But we have lots of results about derivatives, and these
can be "read backwards" to tell us about antiderivatives. The various tricks
that can be used to figure out antiderivatives this way are often referred
to as "techniques of integration",
and they can often be used to get exact results for integrals without having
to go through the tedious limiting process. Once we have an antiderivative,
say F, for the function f, then we can compute any definite integral of
f , say from a to b, by taking the difference F(b)-F(a). (Basically since
the area between x=a and x=b is the area up to b minus that up to a.)

BUT it is not always possible to find an antiderivative. So we always
have to be ready and willing to come back and work with the Riemann Sums
or some other numerical method.

ALSO a knowledge of how integrals correspond to limits of sums is essential
for understanding various applications
of integration to other subjects.

You might also check our 'raw list' (of links provided without comment) to see if there are any more examples there that we haven't yet included here.

If you have come across any good web-based illustrations of these and
related concepts,

please do let
us know and we will add them here.

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