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