Calculus With Measuring Cups

Caroline Labenski                          Bruce Piper 
Merrimack College                          Rensselaer Polytechnic Institute
North Andover, MA                          Troy, NY
E-Mail :           E-Mail :  

Today, computers have become invaluable tools used for solving mathematical problems. Solving such problems demands a complete understanding of the mathematics involved as well as the ability to implement the needed mathematical procedures on the computer. These skills are critical for today's mathematics students as they prepare for their futures. It is with this in mind that we present calculus computer projects, to be used in conjunction with a traditional text, which demonstrate the typical use of a computer as a tool for solving application and theoretical-based problems at the calculus level.

Each of our projects contains a combination of three distinct activities which develop the students' ability to read, write, and apply mathematics. Before detailing each of these activities, we will address our approach to the difficult task of determining the proper topics from which to create projects.

Choosing proper material for a project is critical to its success. We have found that most calculus topics may be used successfully, but some topics lend themselves more readily to the computer environment than others. Topics that are graphical in nature work well on the computer due to their visual payoff for the students. Students, like us all, feel most satisfied when their hard work culminates in a substantial reward. This reward may vary from a graphical image (as in our measuring cup project shown below), to the insight into the inner workings of a complex mathematical idea.

(1) Introduction of a Mathematical Concept

In each of our projects, the students are first introduced to a mathematical concept through a succession of probing questions and explanations. (Typically this concept has previously been presented in lecture.) At this stage, it is the student's goal to gain a solid understanding of the equations and theoretical ideas related to the concept. Having the students concentrate first on the mathematics in a project, before considering an application, serves two distinct purposes. First it insures that the material is fresh in their minds as they proceed to more difficult problems and second it helps them to see the distinction between the mathematical concept and the application of this concept to a problem. (In the classroom however, topics should be motivated via application or theory where appropriate.)

(2) Computer Implementation of Mathematical Procedures

The implementation of a mathematical procedure on the computer, in our case through the software tool Maple, on the surface appears simple, but in reality it demands a complete comprehension of the concepts and notation by the students. The translation of some procedures to the computer simply involves mimicking the steps done by hand using the syntax of the software, but for other procedures, the computer syntax requires that additional or new steps be inserted to "set up" later ones. This latter process can be very confusing and difficult for the students. As a result, this stage is the one that requires the most guidance.

Determining the appropriate amount of guidance, though, can be tricky. We have found that limited guidance which allows the student to make the major contribution towards the solution of the mathematical problem, while guiding them around non-mathematical computer pitfalls, seems to work best for three reasons. First, it gives the students a sense of accomplishment and learning upon completion. Second, any mathematical misconceptions will be exposed and so may be corrected before proceeding. Lastly, the focus always is on the mathematics and how the computer should be used as an aid to the mathematical thought process and not as a replacement for it.

(3) The Application : Development of the plot

Analogous to the way the finale of a good mystery ties together the seemingly unrelated events which occurred throughout the story, the solution to the final problem of a project should depend upon the combination of knowledge obtained from the previous activities and lectures. Few problems one encounters are solvable with one idea or tool; almost universally a combination of tools, knowledge, and abilities are needed. As a result, students should be exposed to such problems as they learn, to enhance their problem solving skills and intuition for determining the proper tools for a situation.

In the final application-based problem of our projects, the students must determine the proper combination of computer, mathematical and reasoning skills needed to solve the problem. Having mastered the computer and mathematical skills in earlier activities, the students are able to focus on the analysis of the problem. Our hope is that through this analysis, the students come to appreciate the power and applicability of mathematics to our every day lives.

Sample Calculus Project: Determining the Volume of a Solid of Revolution

Maple commands needed: fsolve, subs, int, plot, plot3d, display3d

Introduction: As an employee of the Artsy Cookware Company, you have been assigned to join their cookware design team which consists of bakers, artists, and mathematicians. The team's goal is to create a more interesting and aesthetically pleasing set of measuring cups. The artists have drawn initial designs for these cups and now it is your responsibility to create computer images of their work and determine where the measuring marks should be placed using your knowledge of volumes of solids of revolution.


(1) The Wavy Measuring Cup

The artists on your team have designed a non-standard measuring cup with wavy sides which they believe to be quite aesthetically pleasing. All cross-sections of their measuring cup are circular and it has been determined that the cup may be formed by revolving the region bounded by the curves:

x=0, x=1.5, and f(x) = sin(10x)/10 + 3/5 - ((2-x)^4)/32       Equation(1)

about the x-axis.

(a) To get a feel for what this region looks like, enter the function f into Maple and create the two-dimensional region described above by plotting the function f over the domain [0, 1.5] and the vertical the line x=1.5 for 0<=y<=f(1.5) on the same set of axes. (This restriction on y will restrict the line to bound only the region in question. Do you see why?) The line x=0 will be automatically drawn for us as the y - axis.
To visualize the solid formed by rotating this region about the x-axis, we may either visualize it in our heads or use Maple. For the latter, we must first discuss how to plot three-dimensional solids of revolution in Maple.
*** The following would be discussed in class prior to the project assignment. Also note that this project would be assigned prior to the introduction of the three-dimensional coordinate system. **
To create the equation for a three-dimensional solid of revolution formed by rotating a region about the x-axis, we must consider how the boundary surface of this solid is formed. A surface of this type is formed by allowing x to travel through the domain of f and for each value in the domain, having the point (x,f(x)) trace out circles, parallel to the y-z plane, with center (x,0) and radius f(x).

To create these circles, we use the idea of the unit circle in the x-y plane which has every point along it defined by the equation (x,y)=(cos(t),sin(t)) for 0<=t<=2Pi. Since we want cross sectional slices of our surface to be circles parallel to the y-z plane with radius f(x), then for each x in the domain of f, the corresponding cross sectional circle for our surface of revolution is given by (y,z)=(f(x)cos(t),f(x)sin(t)) for 0<=t<=2Pi. Hence the surface of revolution formed by rotating a function y=f(x) about the x-axis is given by:

     (x,y,z) = (x,f(x)cos(t),f(x)sin(t))   (0<=t<=2Pi)      Equation(2)

To define and plot a surface of revolution in Maple, we use the syntax:

	any_name := [ x, f*cos(t), f*sin(t) ]; 
	plot3d( any_name, x = a..b, t = 0..2*Pi ); 
where f is a function defined over the domain [a, b].
(b) In Maple, plot the three-dimensional surface of revolution for the function f defined in equation (1) to form the wavy measuring cup and experiment with the three-dimensional plot window options.
You may have noticed that your measuring cup does not have a base. To define a base for our cup at x=0, we will create a surface that is a solid circle parallel to the y-z plane with radius f(0) and center (0,0).

To create a circle with center at (0, 0) and radius f(0) we use the equation:

	(x,y,z) = (0,f(0)cos(t),f(0)sin(t))	            Equation(3)

for 0<=t<=2Pi. But to create a solid circle, we will need to fill the inner portion of the circle given in equation (3) with smaller circles which have the same center but varying radii from zero to f(0). To define and plot these circles in Maple, we use the syntax:

	base_circles := [0, y*cos(t), y*sin(t) ]; 
	plot3d(base_circles, y = 0..f(0), t = 0..2*Pi);
Notice that y defines the changing radii of the inner circles which fill the outer circle.
(c) Plot the base of the measuring cup in Maple and explain in your own words how the above Maple syntax achieves this figure.
(d) Experiment with varying the radii of the circles to form larger and smaller solid circles. Also create a solid circle with a hole in it.
While you have been creating the cup, the artists have decided that the portions of the cup between each measuring mark and the base should be different colors. As a result, we must now discuss how to plot multiple different color three-dimensional images on one set of axes using the display3d command.

Normally, to plot multiple three-dimensional graphs, say f, g, and h, on one set of axes, we issue the command, plot3d({f, g ,h}, x=a..b, y = c..d); and Maple plots the three graphs on one set of axes assigning random colors to each. To assign a specific color, say blue, to the graph of one three-dimensional function, f(x,y,z), in Maple over the domain [a,b] and [c,d], we adjust the plot3d command to read: plot3d(f, x=a..b, y = c..d, color = blue);.

However, to combine the above two ideas and assign different colors to multiple graphs that are to be plotted on one set of axes, the procedure is more complicated. For this, we use the display3d command. This command requires that we first create plots of each of the graphs in question and assign each plot a distinct color and name. For example, the general syntax for plotting two functions, f and g, on the same set of axes with f in plum and g in coral is:

	fplot := plot3d (f, x = a..b, y = c..d, color = plum): 
	gplot := plot3d (g, x = a..b, y = c..d, color = coral):
	display3d([fplot, gplot]); 
The first command allows you access to particular Maple commands which are stored in the Maple "plot" library. The next two commands of this sequence create the three-dimensional plots of each of the functions in the specified colors and assigns them to the names fplot and gplot respectively. The use of a colon suppresses the output from these lines. Lastly, the display3d command is used to combine the two graphs onto one set of axes and display them on the screen in their assigned colors.
(e) Plot the measuring cup and its base on one set of axes making each a different color.

We now must determine where to place the measure marks along the measuring cup at 3/4 cup, 1/2 cup, 1/4 cup, and 1 cup. Recall that the definition for the volume of a general solid is:

   integral_a^b A(x)dx         Equation(4)

where A(x) gives the cross-sectional area at each value of x, between a and b, along the x-axis. Since our solid is formed by rotating a two-dimensional region about the x-axis, all cross-sections of our solid will be circles. Therefore the area of each cross-section of our solid of revolution is given by the area of each circle, Pi*r^2 = Pi(f(x))^2, f(x) being equal to the radius of each circle since it is the distance between the curve and the x-axis which we are rotating about. Hence the equation to determine the volume of a solid of revolution found by rotating the region bounded by y=f(x) and x=a, and x=b about the x-axis is given by:

    integral_a^b Pi(f(x))^2dx        Equation(5)

(f) Using Maple and the formula in equation (5), determine the volume of the entire measuring cup. Recall that to integrate a function f over the interval [a,b] in Maple, we use the "int" command which has general syntax: int ( f, t = a..b );.
(g) In Maple, using the "int" and "fsolve" commands, determine where along the x-axis to mark one cup. It will be easier if you first create an equation representing volume, then set this volume equation equal to the volume you wish to find and use the "fsolve" command to solve the equation.
(h) Repeat question (g) to determine where along the x-axis the marks for 1/4 cup, 1/2 cup, and 3/4 cup should lie.
(i) Display the final measuring cup according to the artists, wishes. Specifically, display the measuring cup with 6 different colors, one for the base and one for each of the 5 portions divided by the marks for 1/4 cup, 1/2 cup, 3/4 cup, and 1 cup. (The last portion will be from the 1 cup mark to x=1.5.)

(2) The Wavy Cup with a Spoke

This question would now be similar to question (1) resulting in the creation of the measuring cup shown with an inner spoke. The spoke in this cup extends the level of difficulty beyond that of question (1). Its contents, as well as other portions of this project, have been omitted for brevity.

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