When approximating the area under a curve by a sum of rectangles, there are various ways one can choose the heights of the rectangles. In the applet referred to above we are given the choice of using the function value at the right end, left end, or middle of each interval.

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). The little "Flash" animation that's also on our departmental home page illustrates both of these for a particular example, and 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.

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 a rectangle
between the graph and the x-axis would have height -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.

*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.

Although rectangles can always be used, and if thin enough will give a decent approximation, it is often possible to get an accurate result more quickly by making a "less jagged" approximation to the curve. An easy way to do this is by using trapezoids instead of rectangles.

At the
Numerical
Integration Tutorial, created by Joseph
Zachary, from the University of Utah, a JAVA applet makes it possible
to display any one of a number of functions, and to calculate the area
between the function, two moveable white lines, and the *x*-axis.
Various numerical methods of calculation can be used, including the trapezoidal
method.

More links about Numerical
Integration can be found in our "raw list" (of resources that are just
listed without comment).

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 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 function, or conversely the
area 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.

BUT they don't always work. 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.

The connection between the integral (or area) and derivative (or slope)
problems is known as the **Fundamental Theorem
of Calculus**. It is important not just as a tool for quickly
finding some integrals, but also for helping us to recognize the relationship
between processes of change and accumulation in many applied fields. This
theorem is what enabled the rapid advances in our understanding of the
physical world that underlie the industrial revolution. Its discovery is
without a doubt one of the most significant events in all of human history!

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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