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