Objective: The goal of this demo is to provide instructors and students with interactive tools for approximating area under a curve and arclength using elementary numerical methods. Level: Integral calculus courses in high school or college. Prerequisites: Familiarity with integrals, area under a curve, and arclength. Platform: Included are three Flash interactive programs that require the free Flash player. Click here to download and install the Flash player. Instructor's Notes: Most calculus texts present a section on Numerical Integration that illustrates Trapezoidal Approximations and Approximations Using Parabolas which are referred to respectively as the Trapezoidal Rule and Simpson's Rule. For the Trapezoidal Rule, the usual development approximates the integral
by using equispaced partitions of [a, b] of length h > 0 and summing the areas of the resulting trapezoids as shown in Figure 1 where h = 1/2.
The approximation T(h) for the area under the curve for the Trapezoidal Rule is given as
The impression students often get is that equispaced partitions are always used. A small value of h is chosen to obtain accurate approximations over portions of [a, b] where large function variations occur, but the same value of h is used in regions where the curve is nearly linear resulting in more computation than is really necessary. For example see Figure 2.
To permit students to experiment with selecting partitions interactively for Trapezoidal approximations we have have constructed a Flash program that displays the geometric constructions involved and computes the approximate area. Figure 3 shows the program's screen. Initially the program requests that the user input a name (provides an identification) which is then displayed and then we make visible the input boxes for the domain (default is [-10, 10] ) and the input box for the function formula. Once both the domain and function formula are entered click the GRAPH button to display the curve. Coordinate axes will appear when appropriate. A SELECT POINTS button will appear. Clicking it will let you choose points in any order along the curve. In fact the mouse need not be positioned on the curve because a click establishes an x-coordinate and the corresponding y-coordinate is computed to establish a point on the curve. (BEWARE: negative function values should be avoided.) When you are done selecting points click the FINISHED SELECTING button which completes the display of the trapezoids and then computes and displays the sum of the area of the trapezoids.
Click here to execute the Flash program for the Trapezoidal Rule. For Simpson's Rule we have a Flash program that has the same functionality as that for the Trapezoidal Rule but uses parabolic arcs over successive pairs of subintervals of the partition. Since we need pairs of subinterval, an odd number of points must be selected in the partition. The program forces you to choose an odd number of points by not displaying the FINISHED SELECTING button when only an even number have been selected. In this program we display each parabolic arc and sum the areas under the successive arcs to obtain the Simpson's Rule estimate of the area under the curve. Figure 5 illustrates such an approximation for f(x) = x3 -3x2 + 9 over [-1, 3]. The exact value is 28.
Click here to execute the Flash program for the Simpson's Rule. The programs for both Trapezoidal Rule and Simpson's Rule can be used to have students discover properties of these approximation techniques by experimentation with linear and quadratic functions. To approximate the total area bounded by y = f(x) and the axis just plot |f(x)| over the interval of interest. Working with these interactive programs also lays the foundation for more realistic area approximation problems like the following.
The usual development for the length of a plane curve for a continuously differentiable function y = f(x) shows a piece-wise linear approximation like that shown in Figure 7. Then an argument shows that the limit of the sum of the lengths of the line segments as we make the partition length of the largest subinterval in the partition go to zero is the integral
Computation of an antiderivative for the arclength integral is usually not possible except for very elementary functions or for contrived functions that lend themselves to performing the square root. We have a Flash program for approximating the arclength of y = f(x) that determines Riemann sums for the arclength integral. Its functionality is similar to the previously discussed Flash programs. The user enters name, domain, function formula, and then by clicking an appropriate button select points along the graph of f(x) to create a piece-wise linear function like that shown in Figure 7. The program them computes the sum of the lengths of the line segments as an approximation to the arclength. In Figure 8 is an example for whose arclength is approximately 11.3208.
Click here to execute the Flash program for approximating arclength. A sample set of numerical exercises that incorporates all three Flash programs is available as a pdf file by clicking here. One of the exercises asks for the approximation of the arclength of a corrugated sheet in order to determine the width of a plain sheet needed for a stamping machine to create the corrugated sheet. This was used by several calculus sections in the spring term of 2008 at Temple University. To download a zipped file containing all three flash programs click here. Extract the files into a folder. You must ensure that the file AC_RunActiveContent.js (which is part of the download) is in the folder containing the Flash files. To execute the Flash programs click on the corresponding html file which is included. Click here to download and install the free Flash player. Credits: This demo and the Flash programs files were developed by David
R. Hill
and is included in Demos
with Positive Impact with his permission.
The Flash programs adapted components from the Flash |
