Volumes of Revolution Using Cylindrical Shells

Manipulate[
a=0;
b=2;
dx=((b-a)/n);
f[x_]:=2(x^2)-(x^3);
startpos[1,k_]:=a+(k-1)dx;
startpos[2,k_]:=a+k*dx;
startpos[3,k_]:=a+((2k-1)/2)dx;
curve=ParametricPlot3D[{x,0,2(x^2)-(x^3)},{x,a,b},PlotRange->{{-b,b},{0,b},{-b,b}}];
end=Graphics3D[{Table[Cuboid[{{a+(l-1)dx,0,0},{a+l*dx,+.01,f[startpos[loc,l]]+.01}}],{l,1,n,1}]}];
If[sho==1,
solid=ParametricPlot3D[{x*Cos[y],x*Sin[y],2(x^2)-(x^3)},{x,a,b},{y,0,θ-.01},PlotStyle->{Blue,Opacity[0.25]},PlotRange->{{-b,b},{0,b},{-b,b}}],
solid=Graphics3D[{Yellow,Opacity[0.5],Table[Cylinder[{{0,0,0},{0,0,f[startpos[loc,i]]}},i*dx],{i,1,n,1}]} ]];
Show[{curve, solid, end},
AxesLabel->{Style["x",Italic],Style["y",Italic], Style["z",Italic]},AxesOrigin->{0,0,0},ImageSize->{400,400}],
{{loc,1,"location"},{1->"left side",2->"right side", 3->"center"}},
{{sho,1,"show"},{1->"solid",2->"cylinders"}},
{{n,2,"number of segments"},2,15,1,Appearance->"Labeled"},
{{θ,0,"solid sweep angle"},0,π,-.1,Appearance->"Labeled",Enabled->(sho===1)},
AutorunSequencing->{1,2,3},
TrackedSymbols:>{θ,n,loc,sho}]

CAPTION

Calculus offers two methods of computing volumes of solids of revolution obtained by revolving a plane region about an axis. These are commonly referred to as the disc/washer method and the method of cylindrical shells, which is shown in this Demonstration. In this example the first quadrant region bounded by the functionf(x)=2(x^2)-(x^3) and the x axis is rotated about the y axis. You can see the solid of revolution or an approximation by a set of cylindrical shells. As the number of shells is increased you can see that the approximation becomes closer to the solid. You can approximate the volume using shells whose heights are given by the function value at the left, right, or center of the x axis interval that generates the shell.

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Volumes of Revolution Using Cylindrical Shells" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/VolumesOfRevolutionUsingCylindricalShells/

Contributed by: Stephen Wilkerson (Towson University)

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