A collection of tutorials (Flash and Java) on various applications of integration including area between two curves, volumes of solids of revolution, arc length, average value of a function, work, and moments and centers of mass. This instruction comes in many forms: text, graphics, animations, visualizations, practice exercises with solutions, etc. In addition to Macromedia Flash and Java, the tutorials employ a number of computer programs including: LiveMath (MathView), Microcalc, JKGraph and MPP3D (a plotting program). Most of these programs, both stand-alone and plug-in types, are free downloads. The site provides numerous examples with detailed instructions for using the TI-85 and -86 graphing calculators. One of the topics (Area between two curves) also includes practice exercises with solutions.
Minimally, this site requires a Java-enabled browser, preferably Netscape, and
browser plug-ins for LiveMath and Macromedia Flash. Other computer programs are
referenced and employed, but no plug-ins are needed.
Identify Major Learning Goals:
To provide tutorials on applications of integration.
Target Student Population:
Students taking a course in single-variable calculus.
Prerequisite Knowledge or Skills:
Knowledge of integral calculus
The applets are extremely well-designed and implemented and provide step-by-step solutions for various application exercises. The use of color in the tutorials and the animations and visualizations are excellent and would certainly help students come to an understanding of the mathematical concepts. Some of the computer programs used to generate visualizations are available as downloads from the UTK Math Archives site; while these programs are a bit dated, they may still be of value to an interested user.
The only error found on the site involved an animation and graph for finding volumes of solids of revolution by the cylindrical shell method. The animation and graph did not correctly correspond with the function under consideration.
Potential Effectiveness as a Teaching Tool
The materials are excellent and the use of color in the tutorials as well as the animations and visualizations would certainly help students come to an understanding of the mathematical concepts. The animation and discussion concerning volumes of solids of revolution that show how increasing the value for n, the number of subintervals, leads to a reasonable estimate of the volume is quite effective in showing students the nature of the limit process. In addition, the visualization of the slab process for finding volumes was also helpful and would serve as an excellent classroom demo. A nice feature for self-study is that each section included a link to learning objectives.
There were a few places where a bit of additional explanation would have been helpful, especially for students who are pursuing self-study. For examples, in finding the area for parametric curves, the reason for the different limits of integration for x and t could have been elaborated upon. When finding centers of mass, the expression for the moment around the x-axis might have been expressed both as a difference of squares and as the product of a sum and a difference of f and g in order to more clearly make the connection between the two forms. While optional, many of the sections contain software tutorials. The volumes section uses the package Microcalc extensively which does not seem to be readily available anymore. This diminishes the effectiveness of that section slightly.
Ease of Use for Both Students and Faculty
The applets were very easy to use and the site as a whole was well-laid out with a Table of Contents. Each screen included a Help link and a Back button that returned the user to the Table of Contents.
There was no Previous button on the applets in case a student wanted to review the previous screen; the Back button on the browser returned the user to the Table of Contents and the applet had to be restarted from the beginning. Another concern is that many of the links on the Help section are broken.
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