Matthew Lehrman

Footnote 18

Mat 271 12:40

A Further Examination of the Volume

of Revolution Function in Maple

The maple function for rendering the volumes of cylindrical shells is an extremely useful tool. Although most of the problems can be a calculated and drawn by hand, often it is difficult to overcome the problem of representing a three dimensional shape in two dimensions. This is where the ability to render the volume with Maple in real time and manipulate the angle from which they are viewed comes into play. While still a two dimensional representation, the computer gives the student the power to explore the shape, rotating or zooming in and out at will. However, before any graphs can be rendered, it is necessary to call the “with(Student[Calculus1])” function. Without this command, the “VolumeOfRevolution” will only give error messages. A successful rendering can be seen below in problem six from section 6.3.

> VolumeOfRevolution(3+2*x-x^2,3-x,x=0..3,output=plot,axis=vertical);

> VolumeOfRevolution(3+2*x-x^2,3-x,x=0..3,axis=vertical);

The base call function of VolumeOfRevolution(); is set up so that the outside curve is listed first, followed by the inside curve, limits, graphical representation (optional) and the orientation about the axis. Maple clearly distinguishes between the inside (magenta) and outside (red) curves with different colors. The colors are also semi-transparent and darken in areas that would be invisible if the colors were solid. These two features work to give a better illusion of depth, an effect that is very difficult to achieve by hand. The particular example above happened to be centered about the vertical axis (y), so the “axis=vertical” had to be added in order to obtain the correct graph. An example of a problem not centered about the vertical axis was number eleven (below).

> VolumeOfRevolution(8,x^(3),x=0..2,output=plot);

> VolumeOfRevolution(8,x^(3),x=0..2);

Since the default of the program is to generate a graph about the horizontal (x) axis, there is no need to add any extra conditions.

An additional bonus of using maple is the ability to quickly calculate the volume of the cylindrical shells by simply removing the “output=plot”. In both examples, the volume is displayed in blue. This allows the user to check answers calculated by hand if solutions are not provided.

However, there are limitations to what the program can represent. There are some graphs which are not able to be displayed properly, such as problem 15 (below).

>VolumeOfRevolution((x)^(2),0,x=1..2,output=plot,axis=vertical);

> VolumeOfRevolution((x)^(2),0,x=1..2,axis=vertical);

The outer shell is left only partially rendered and an inaccurate shape is represented. It is then left to the imagination of the user to add a wall around the center and the outside rim. Fortunately, the numerical value of the volume is still calculated correctly.