Objectives Defined:

The purpose of this module is to improve the student's conceptual and computational understanding of how a solid can be generated through the construction of disks and shells. Furthermore, the volumes of various solids will be determined through those two constructions using traditional calculus approaches. These goals will be achieved by

1. Reviewing the construction of disks/washers by revolving a representative rectangle.

2. Reviewing the construction of shells by revolving a representative rectangle.

3. Exploring each method about various axes of revolution.

4. Examining the notation involved in each process.

5. Providing student problem sets.

Review of Disk/Washer Construction Method:(partly animated)

As with the Riemann Sum Construction for determining the area beneath a curve, we can create the same representative rectangular strip and use it to generate 3-D solids. The function f(x) = graphed on [0,2] shown below includes one such representative rectangle.

When we revolve the rectangle about the x-axis, then a circular disk is formed.

Animation of a Revolving Disk

The animation below shows how when we revolve one rectangle a corresponding disk is created.

To run the animation "point and click on the figure". You can control the animation with the Animation Toolbar which appears at the top.

The volume of this lone disk is given by:

where f(x*) is an approximate radius for the disk                                    

As we let x* vary from 0 to 2, we can create a volume of disks.

Animation of a Solid Filled With Disks

The animation below shows how the repeated usage of disks can create a solid of revolution. Once agian, "point and click" on the picture and then use the Animation Toolbar to control the animation.

Summing up the volumes of the individual disks and letting our thickness be a limiting process, we obtain the integral notation for the volume of the solid of revolution formed. For our problem the solution is given by.

>	Radius(x):=x^(1/2): Integrand:=Pi*Radius(x)^2: Volume:=Int(Integrand,x=0..2)=int(Integrand,x=0..2)*`  cubic units`;

The general formula is given below.

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The Volume Formula for Disks/Washers Rotated about the X-Axis

In words, the volume that is generated is the integral whose integrand is equal to pi times the radius squared where the radius is measured from the x-axis to the curve f(x). If our rectangular strip is contained between two functions f and g then we create a washer which has an "inner" and "outer" radius. We must adjust our formula accordingly. See the diagram and the accompanying formula.

The solution to this scenario is determined by the code below. Simply place the cursor anywhere in the red code and press ENTER.

>	Outer(x):=x^(1/2): Inner(x):=(1/2)*x: Integrand:=Pi*(Outer(x)^2-Inner(x)^2): Volume:=Int(Integrand,x=0..2)=int(Integrand,x=0..2)*`  cubic units`;

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The Volume Formula for Disks/Washers Rotated about the Y-Axis

The procedure and formula for rotating about the y-axis is analogous except that our radii must be in terms of y and our limits of integration run along the y-axis. Our volume formula in this case reads.

note: the functions f*(y) and g*(y) represent new functions whose independent variables are y.

Review of Shell Construction Method:

The function f(x) = graphed on [0,2] shown below displays a representative rectangle which is parallel to the x-axis. If we rotate the rectangle about the x-axis then a "shell" is created.

When we revolve the rectangle about the x-axis, then a circular shell is formed which has thickness

The volume of this lone shell is given by:

As we let y vary from 0 to sqrt(2), we can get as many shells as we like creating an approximation to the true volume that would be generated.The animation below creates the generated solid through multiple shells. As before, use the Animation Toolbar after clicking on the figure to control the animation.

Animation Showing Shells Creating A Solid

Summing up the volumes of the individual disks and letting our thicknesses be a limiting process we obtain the integral notation for the volume of the solid formed. For our problem the solution is given by.

>	Radius(y):=y: Circum(y):=2*Pi*Radius(y): Length(y):=2-y^2: Integrand:=Circum(y)*Length(y): Volume:=Int(Integrand,y=0..sqrt(2))=int(Integrand,y=0..sqrt(2))*`  cubic units`;

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The Volume Formula for Shells Rotated about the X-Axis

In words the volume is aquired by summing up the volumes of an infinite number of nested shells whose radii vary with y and whose heights(lengths) vary according to the curve.

Similarily, if we rotate a strip around the y-axis we obtain the following:

       Our volume formula changes as follows:

The Volume Formula for Disks/Washers Rotated about the Y-Axis

Further Examples:

The figure below shows a partially filled half-circle. If we rotate this figure we will get a hemisphere which is partially filled.

The intagral to determine the volume generated is given by (disk method);

This simple MAPLE code will evaluate this integral. Place your cursor in the red restart word and press ENTER.

>	restart: Volume:=int(Pi*(r^2-y^2),y=-r..h-r);

The answer can be written cleanly as:

You should verify that algebraically.

Major Source Code:

Within this section are the coding routines to produce the images conatined in this lab. The /*Creates... prompt describes the routines purpose.

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restart:with(plots):/*Creates a vertical rectangle. h:=PLOT(POLYGONS([[0.8,0.4],[0.9,0.4],[0.9,sqrt(0.9)],[0.8,sqrt(0.9)]],COLOR(HUE,0.8)),TEXT([0.85,-0.50],'delta_x_thick',COLOR(HUE,5.9)),     AXESSTYLE(NORMAL),VIEW(0..2,-1..1.5),AXESTICKS(4,4)): g:=plot({0.5*x,sqrt(x)},x=0..2,thickness=3,color=[gray,brown]): k:=textplot({[1.1,1.4,'f(x)=sqrt(x)'],[1.3,0.35,'g(x)=x/2']}): display(h,g,k);

Warning, the name changecoords has been redefined

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restart:with(plots):with(plottools):/*Creates a vertical Washer.disk1:=(a,b,c,r,h)->rotate(cylinder([a,b,c],r,h,color=magenta),0,-Pi/2,0):disk2:=(a,b,c,r,h)->rotate(cylinder([a,b,c],r,h,color=white),0,-Pi/2,0): myfunction1:=(x,y)->plot3d(sqrt(x),x=0..2,y=0..2,thickness=3,color=brown): myfunction2:=(x,y)->plot3d((1/2)*x,x=0..2,y=0..2,thickness=3): display({disk1(0,0,0.9,1,0.1),disk2(0,0,0.91,0.5,0.1),myfunction1(x,y),myfunction2(x,y)},scaling=unconstrained,style=patchnogrid,axes=normal,orientation=[-80,90], title="Rotated Rectangle\nCreates a Washer",titlefont=[TIMES,ROMAN,14],lightmodel=light4,TEXT([0.6,1,1.4],`f(x)=sqrt(x)`),TEXT([1.6,1,0.4],`g(x)=0.5x`));

Warning, the name changecoords has been redefined

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restart:with(plots):with(plottools):/*Creates a sequence of Disks. washer:=(a,b,c,r,h)->rotate(cylinder([a,b,c],r,h),0,-Pi/2,0): Disks:=(f,a,b,n)->display(seq(washer(0,0,a+k*(b-a)/n,f(a+k*(b-a)/n),(b-a)/n),k=0..n),scaling=unconstrained,style=patchnogrid,axes=normal,lightmodel=light2,orientation=[-83,90], title="Multiple Disks",titlefont=[TIMES,ROMAN,14]): Disks(sqrt,0,2,15);

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restart:with(plots):/*Animates one revolving Disk. f:=y^2: F:=unapply(f,y): Frame:=1: for n from 1 to 64 do Graph:=plot3d([f,y*cos(t),y*sin(t)],y=0..sqrt(2),t=0..-2*Pi,color=khaki): rotaxis:=pointplot3d([[0,0,0],[2,0,0]],style=line,color=green,thickness=6): disks:=seq(pointplot3d([[F(1),0,0],[F(1),(1)*sin(k*Pi/32),(1)*cos(k*Pi/32)]],style=line,thickness=10,color=blue),k=1..n): pic||Frame:=display(Graph,rotaxis,disks): Frame:=Frame+1:od: display(pic||(1..Frame-1),insequence=true,style=wireframe,orientation=[-82,93],axes=normal);

Warning, the name changecoords has been redefined

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restart:with(plots):with(plottools):/*Animates the filling of a solid with Disks. radi:=x->x^(1/2): First:=NULL: Frame:=1: for k from 1 to 21 do Cur:=tubeplot([x,0,0,x=0..2],radius=radi(x),color=khaki,style=wireframe,tubepoints=24): rotaxis:=pointplot3d([[0,0,0],[2,0,0]],style=line,color=green,thickness=6): L:=line([(k-1)/10,0,0],[(k-1)/10,((k-1)/10)^(1/2),0],thickness=9): disk:=seq(rotate(L,n*Pi/32,[[0,0,0],[2,0,0]]),n=0..64): First:=First,disk: gr||Frame:=display(Cur,rotaxis,First,color=blue): Frame:=Frame+1:od: display(gr||(1..Frame-1),insequence=true,orientation = [-82,93]);

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restart:with(plots):/*Creates a horizontal strip. h:=PLOT(POLYGONS([[0.8,sqrt(0.7)],[2,sqrt(0.7)],[2,sqrt(0.9)],[0.8,sqrt(0.9)]],COLOR(HUE,0.8)),TEXT([0.4,0.9],'delta_y_thick',COLOR(HUE,5.9)),     AXESSTYLE(NORMAL),VIEW(0..2,-1..1.5),AXESTICKS(4,4)): g:=plot(sqrt(x),x=0..2,thickness=3,color=[brown]): k:=textplot([1.1,1.4,'f(x)=sqrt(x)']): display(h,g,k,title="Horizontal Rectangle\nfor Shell Method",titlefont=[TIMES,BOLD,14]);

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restart:with(plots):with(plottools):/*Creates a Shell. disk1:=(a,b,c,r,h)->rotate(cylinder([a,b,c],r,h,color=magenta),0,-Pi/2,0):disk2:=(a,b,c,r,h)->rotate(cylinder([a,b,c],r,h,color=white),0,-Pi/2,0): myfunction1:=(x,y)->plot3d(sqrt(x),x=0..2,y=0..0.2,thickness=8,color=brown):test:=rotate(plot3d(x^(1/2),x=0..2,y=0..sqrt(2)),Pi,1,0): myfunction2:=(x,y)->plot3d(-sqrt(x),x=0..2,y=0..0.2,thickness=8,color=khaki,style=wireframe): display({disk1(0,0,0.9,1,1),disk2(0,0,0.91,0.8,1.001),myfunction1(x,y),myfunction2(x,y)},scaling=unconstrained,style=patchnogrid,axes=normal,orientation=[-71,90], title="Rotated Rectangle\nCreates a Shell",titlefont=[TIMES,ROMAN,14],lightmodel=light4,TEXT([0.6,1,1.5],`f(x)=sqrt(x)`));

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with(plottools):/*Animates the creation of one Shell. r[0]:=POLYGONS([[0,0.8,0],[0,0.9,0.0],[0,0.9,sqrt(0.9)],[0,0.8,sqrt(0.9)]],COLOR(HUE,0.8)): l := [[0,0,0],[0,0,1]]: a := Pi/50: c := 1: while evalf(a-2*Pi) < 0 do  r[c] := rotate(r[0],a,l);  a := a+Pi/50;  c := c+1; end do: display([seq(r[i],i=0..c-1)],title="Shell Construction\nY-Axis Rotation\nMultiple Views", titlefont=[TIMES,BOLD,14],axes=normal,view=[-1..1,-1..1,0..2],orientation=[5,66]);

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restart:with(plots):with(plottools): /*Animates filling with Shells. radi:=y->y^2: height:=x->sqrt(x): First:=NULL: Frame:=1: param1:=blue: param2:=red: for k from 1 to 10 do if irem(k,2)=0   then     param:=param2   else     param:=param1 fi: Cur:=tubeplot([x,0,0,x=0..2],radius=height(x),color=khaki,style=wireframe,tubepoints=24): rotaxis:=pointplot3d([[0,0,0],[2,0,0]],style=line,color=green,thickness=6): L:=line([2*(k-1)/9+.01,sqrt(2)*((k-1)/9)^(1/2),0],[2,sqrt(2)*((k-1)/9)^(1/2),0],thickness=3,color=param): disk:=seq(rotate(L,n*Pi/32,[[0,0,0],[2,0,0]]),n=0..64): First:=First,disk: gr||Frame:=display(Cur,rotaxis,First): Frame:=Frame+1:od: display(gr||(1..Frame-1),insequence=true,orientation = [-76,83],axes=normal);

Warning, the name changecoords has been redefined

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