       RE: Revolving a Rectangle About an Axis

• To: mathgroup at smc.vnet.net
• Subject: [mg32213] RE: [mg32206] Revolving a Rectangle About an Axis
• From: "David Park" <djmp at earthlink.net>
• Date: Sun, 6 Jan 2002 03:38:29 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Ryan,

I think I understand the disk method for finding the volume contained by
surfaces of revolution but I don't understand how rectangles come into it.
What you have is circular disks of different radii that slice the volume of
revolution.

There are a number of ways to make an illustrative plot. Here I show how to
do it using the DrawGraphics package from my web site. It is very easy to
color, use wireframe and combine 3D surfaces with DrawGraphics and more
difficult to do it with standard methods.

I am going to illustrate the disk method for the surface of revolution y =
Sqrt[x] between x = 0 and x = 4, revolved about the x-axis. I will make an
animation showing the surface and the disk sliding along the x-axis.

Needs["DrawGraphics`DrawingMaster`"]

surface[x_, t_] = {x, Sqrt[x]*Cos[t], Sqrt[x]*Sin[t]};
disk[x_][r_, t_] = {x, r Cos[t], r Sin[t]};

The following code defines one frame of the animation.
1) I draw the back half of the surface of revolution in CadetBlue on the
front side and Salmon on the back side.
2) I draw the front half of the surface of revolution using a LightGray
wireframe.
3) I draw the disk representing one slice of the volume in Melon.
4)The NeutralLighting command (which generates a sequence of Options) is
used so that the colors will show well and not be overwhelmed by the default
lighting.

diskframe[xval_] :=
Draw3DItems[
ParametricDraw3D[Evaluate[surface[x, t]],
{x, 0, 4}, {t, -Pi/2, Pi/2}, PlotPoints -> {15, 30}],
LightGray,
UseWireFrame[ParametricDraw3D[Evaluate[surface[x, t]], {x, 0, 4},
{t, Pi/2, 3*(Pi/2)}, PlotPoints -> {15, 30}]],
SurfaceColor[Melon],
ParametricDraw3D[Evaluate[disk[xval][r, t]],
{r, 0, Sqrt[xval]}, {t, 0, 2*Pi}, PlotPoints -> {8, 30}]},
Axes -> True,
NeutralLighting[0.3, 0.5, 0],
PlotRange -> {{-0.2, 5}, {-2.2, 2.2}, {-2.2, 2.2}},
ImageSize -> 500];

This produces the animation.

Animate[diskframe[s], {s, 0.1, 4}]
SelectionMove[EvaluationNotebook[], All, GeneratedCell]
FrontEndTokenExecute["OpenCloseGroup"]
FrontEndTokenExecute["SelectionAnimate"]

I leave the actual integration to the student.

David Park

> From: Ryan R. Rosario [mailto:rrosario11 at hotmail.com]
To: mathgroup at smc.vnet.net
>
> Happy New Year everyone!
>
> I am trying to help a calculus 1 student understand the disk method for
> solids of revolution by using Mathematica graphics. I have created a
> function that produces a series of "typical rectangles" for a
> given function
> and its endpoints.
>
> How can I revolve a rectangle about an axis? I would like to revolve the
> rectangle so that it represents a three dimensional disk. Is this
> possible?
> If so, how can I do it?
>