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 "delta-x") is the width of the rectangle (a difference 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
please do let us know and we will add them here.
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