Note: There are many examples of good visualization utilities for
Riemann Sum approximations that are suitable for first term Calculus. Some
routines display the approximating rectangles together with the sum. This demo
incorporates another visualization aspect: specifically, it
combines the visualization of approximating rectangles together with a graph of the
approximate areas as a function of the number of
rectangles. Thus, the limiting behavior of the approximating sums can be observed
on the graph.
Objective: The basis for
determination of area under a curve and areas between two curves is the
successive approximation of the area using Riemann sums over an appropriate
partition of an interval . The actual area is determined by taking
the limit of Riemann sums as the number of rectangles increases without bound in
such a way that the norm of the partition approaches zero. The purpose of
this demo is to graphically illustrate Riemann sum approximations of areas
between two curves and the limiting behavior of the approximation. This
demo can be used to introduce the idea of area between two curves.
Level: This demo can be presented
in first term Calculus.
Prerequisites: Students
should be familiar with Riemann sums and the idea of approximating rectangles.
The concepts of upper and lower sums should have been introduced. In
addition, it would be useful if students have been introduced to graphical
notions of limits and limiting behavior.
Platform: Any computer system
with MATLAB, v5 Release 11 and the Symbolic Math Toolbox OR the Student Edition
of MATLAB v5.3.
The MATLAB m-file arears.m provides a demonstration of
area approximations.
The MATLAB script prompts the user for two nonconstant
input functions, an interval, and the maximum number of rectangles, nmax.
If functions of x, the partition of the interval [a,b]
is
a=x0, x1,...,xn-1,xn=b
with increments given by
.
In
this case, the area between graphs of ftop(x) and fbottom(x)
is approximated using vertical rectangles (see Figure 1) with height determined using left hand
endpoints of the partition subintervals.
Area of Approximating Rectangle:
Figure 1.Vertical
Rectangles using left hand endpoints.
If functions of y, the partition of the y-interval [a,b] is
a=y0,
y1,...,yn-1,yn=b
with increments given by
.
In
this case, the area between graphs of fright(y) and fleft(y)
is
approximated using horizontal rectangles (see Figure 2) with length determined using lower
endpoints of the partition subintervals.
Area of Approximating Rectangle:
Figure 2. Horizontal
rectangles using lower endpoints.
In either case, approximation begins
with 2 rectangles and increases to nmax in increments of 2.
The graphics display is divided into two subplots (see
animation in title display above). The left hand plot illustrates the approximating rectangles.
The right hand plot displays the numerical values of the successive approximations. As the number of
rectangles is increased, the approximations approach a limiting value which can
be computed using definite integrals, provided the antiderivative can be found.
Instructor's Notes: The area between two curves
can be approximated using Riemann sums. For example, the area between the
graphs of y = x2 and y = 1-x2
on the interval can be
approximated using vertical rectangles as in Figure 3 below.
The area of a representative
approximating rectangle is:
where xi-1 is the left
hand endpoint in the ith
subinterval of .
Note that as the rectangles track across the
region, the top of the rectangle is always on the "top"
graph, y = 1- x2.
That is not always the case as Figure 4 shows.
Figure 3.
The area between the
graphs of y = x2 and y = 1-x2
in the interval can be also be approximated using vertical rectangles,
however, at a point on the interval, the graphs of the functions cross each
other.
Note that when the functions
cross each other, the "top" and "bottom"
functions switch at the point of intersection.
We can use Riemann sums to approximate the
area nonetheless.
Figure 4.
There is nothing sacred about approximating
area between two curves using vertical
rectangles. It is also possible (and sometimes necessary) to approximate using horizontal
rectangles. Consider the area bounded by the graphs of
and on the interval [0, 2], shown
in Figure 5.
The area of a representative
approximating rectangle is given by
where yi-1 is the lower
endpoint in the ith
subinterval of [0, 2]. Note that [0,2] this is an interval of y
values.
Figure 5.
It is important to observe that as the number of subintervals
increases, the area approximation approaches a limiting value.
The MATLAB m-file arears.m generates a graph that
illustrates the approximating rectangles as well as a graph that illustrates the
limiting behavior of the approximations. Click the picture below to
enlarge.
The MATLAB M-file arears.m
can be used with a variety of functions. However, this M-file was designed
to be an instructional tool. It is important that you choose examples
carefully for maximum benefit. Use nmax between 10 and 50, inclusive.
Below are
links to several Riemann Sum routines available for various platforms. The
links listed below will direct you to web pages from which you may obtain
additional information about various routines and author contact
information. Commercially available products are denoted by *.
The links are provided below with the understanding that these routines are the
property of the owners/publishers and should be used only
with their permission.
University of Tennessee: Visual
Calculus
A collection of visualization and computational tools, including interactive
modules, for Riemann Sums. Includes demos using TI-85 and TI-86
calculators, MathView, Maple, and other software packages. Visual
Calculus was authored by Larry Husch and is hosted by Math
Archives.http://archives.math.utk.edu/visual.calculus/4/
University of Arizona:Integral An MSDOS application by Clark Benson and David Lovelock that allows computation and visualization of Riemann Sums http://archives.math.utk.edu/software/msdos/calculus/integral/.html
Animating Calculus*, Ed
Packel and Stan Wagon. A collection of 22 labs utilizing
animations. One of these labs deals with investigation of the definite
integral using Riemann Sums. (Mathematica)
Additionally, we offer a MATLAB Riemann Sum
Routine described below.
MATLAB Riemann Sum
Routine: This routine requires MATLAB v5 R11 and the Symbolic Math
Toolbox OR the Student Version of MATLAB v5.3. Use the slider to increase
the number of approximating rectangles.
When executed, the user may select 3
"demo" functions or input a different function. Click on the picture
below for a larger view of the graphics.
.
.
can be
approximated using vertical rectangles as in Figure 3 below.
can be also be approximated using vertical rectangles,
however, at a point on the interval, the graphs of the functions cross each
other.
and 
on the interval [0, 2], shown
in Figure 5.
