Langara College - Department of Mathematics and Statistics

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Internet Resources for the Calculus Student - Topics in Calculus

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Numerical Integration

The objective of Numerical Integration is to obtain approximate numerical
values for integrals where the corresponding antiderivative can not be
found and without exactly evaluating the limit

of Riemann sums.
Since a Definite Integral is defined as a
limit of Riemann sums, it follows that the value of the integral can be
well approximated by such sums so long as the interval widths are small
enough. 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.

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).

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.