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Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
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Example 5.2.4
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If is the plane region bounded by the -axis and the graphs of and , use the method of disks to calculate the volume of the solid of revolution formed when is rotated about the line .
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Solution
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Mathematical Solution
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Figure 5.2.4(a) shows the region , the axis of rotation, and the inner and outer radii and , respectively.
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use plots,VectorCalculus in
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local q1,q2,q3,q4,q5,q6,q7,q8,q9;
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q1 := plot(x^2,x=0..1,filled=true,color=red, view=[0..4,0..1],transparency=.5, tickmarks=[3,2], labels=[x,y]);
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q2 := RootedVector(root=[2,0],<0,1>);
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q3 := PlotVector(q2,color=black);
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q4 := RootedVector(root=[2,.2],<-1.51,0>):
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q5 := PlotVector(q4,color=blue):
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q6 := RootedVector(root=[2,.7],<-1,0>):
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q7 := PlotVector(q6,color=green):
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q8 := textplot({[1.5,.37,typeset(R)],[1.5,.86,typeset(r)]}):
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q9 := display([q1,q3,q5,q7,q8],scaling=constrained):
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Figure 5.2.4(a) Region
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use plots, plottools,VectorCalculus in
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local q,F,Q,A,p,V,k,spin;
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q := plot(x^2,x=0..1,filled=true);
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F := transform(proc (x,y) options operator, arrow; [x,0,y] end proc);
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V:=RootedVector(root=[2,0,0],[0,0,1]):
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A:=PlotVector(V,color=black,width=.1):
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p:=display([seq(rotate(Q,(2*Pi*(1/30))*k,[[2,0,0],[2,0,1]]),k=0..29)], insequence = true);
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spin := display([A, p], labels = [x,z,y], axes = frame, scaling = constrained,orientation=[-70,72], tickmarks=[3,[0],2]);
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Figure 5.2.4(b) Region rotated
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use VectorCalculus,plots, plottools in
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local q,F,Q,V,A,p,spin,p4,p5,k;
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q:=plot(x^2,x=0..1,filled=[color=red],color=red):
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F:=transform((x,y) ->[x,0,y]):
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V:=RootedVector(root=[2,0,0],[0,0,1]):
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A:=PlotVector(V,color=black,width=.1):
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p:=display([seq(rotate(Q,(2*Pi*(1/30))*k,[[2,0,0],[2,0,1]]),k=0..10)], insequence=true):
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p4:=translate(animate(plot3d,[[r,t,(r-2)^2],r=1..2,t=-Pi..x,coords=cylindrical,filled,style=surface,color=yellow,lightmodel=light4],x=-Pi..Pi, paraminfo=false,frames=11),2,0,0):
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p5:=display([p4,spin],labels=[x,z,y],tickmarks=[4,[0],2], orientation=[-55,70], axes=frame,scaling=constrained);
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Figure 5.2.4(c) Solid generated
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Figures 5.2.4(b) and 5.2.4(c) are animations, the first showing the region rotated about the axis ; the second, the solid evolving from the rotation of region about the line .
Figures 5.2.4(d-f) show how the solid of revolution (Figure 5.2.4(f)) is "constructed" by removing from the solid generated by rotating about the line (Figure 5.2.4(d)), the cylinder of radius 1 generated by rotating about the same axis (Figure 5.2.4(e)).
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use plots, plottools in
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local p1,F,G,p2,p3,p4,p5,p6;
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p1 := disk([0,0],1,color=green):
p6 := disk([0,0],2,color=green):
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F := transform((x,y)->[x,y,1]):
G := transform((x,y)->[x,y,-.02]):
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p2 := plot3d([r,t,(r-2)^2],r=1..2,t=0..2*Pi,coords=cylindrical,style=surface, color=red):
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p3 := display([F(p1),p2,G(p6)]);
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p4 := translate(p3,2,0,0);
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p5 := display(p4, axes=frame, tickmarks=[3,[0],[0,1]], labels=[x,z,y], orientation=[-50,70], scaling=constrained);
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Figure 5.2.4(d) Rotate
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use plots, plottools in
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p3 := plot3d([1,t,z],t=0..2*Pi,z=0..1,coords=cylindrical, style=surface, color=gold);
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p4 := translate(p3,2,0,0);
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p5 := disk([2,0],1,color=green);
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f := transform((x,y)->[x,y,1]);
g := transform((x,y)->[x,y,0]):
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p6 := display([f(p5),p4,g(p5)], scaling=constrained, axes=frame, labels=[x,z,y], tickmarks=[3,[0],2], orientation=[-50,70], view=[0..4,-2..2,0..1]);
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Figure 5.2.4(e) Rotate
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use plots, plottools, Student[Calculus1] in
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p6 := VolumeOfRevolution(x^2,0, 0..1, 'axis'='vertical', 'distancefromaxis' = 2, 'output'='plot',title="",volumeoptions=[color=red,transparency=0],caption="",lineoptions=[color=black]):
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p7 := plot3d([.99,t,z],t=0..2*Pi,z=0..1,coords=cylindrical,style=surface,color=gold):
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p8 := display([p6,translate(p7,2,0,0)]);
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p9 := display(p8, view=0..1,scaling=constrained,axes=frame, labels=[x,z,y],orientation=[-125,70,0],tickmarks=[3,[0],2]);
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Figure 5.2.4(f) Composite solid
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Figure 5.2.4(g) shows the solid in Figure 5.2.4(f) sliced into a stack of disks, each of which becomes a washer because of the central "hole" through the solid. Figure 5.2.4(h) shows a single washer, of thickness (height) , and inner and outer radii and , respectively.
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use Student[Calculus1], plots, plottools in
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p1 := VolumeOfRevolution(2-sqrt(x),1,x=0..1,output=plot,title="", showsum=true,showvolume=false,axes=none, partition=6,method=left,axis=horizontal, showfunction=false,scaling=constrained, distancefromaxis=0,caption="",sumvolumeoptions=[transparency=0,color=brown]):
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p2 := display(rotate(p1,0,Pi/2,0),orientation=[-90,50]);
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Figure 5.2.4(g) Stack of washers
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use Student[Calculus1], plots, plottools in
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p1 := VolumeOfRevolution(1,sqrt(x),x=0..1/2,output=plot,title="", showsum=true,showvolume=false,axes=none, partition=1,method=midpoint,axis=horizontal,showfunction=false,scaling=constrained,caption="",sumvolumeoptions=[transparency=0,color=brown],lineoptions=[color=black]):
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p2 := display(rotate(p1,0,Pi/2,0),orientation=[-90,55]);
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Figure 5.2.4(h) Single washer
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As per Table 5.2.1, the volume of the solid is obtained by the following calculation.
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Maple Solution
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For rotation about a vertical axis, the
tutor provides only the method of shells.
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Nevertheless, Figure 5.2.4(d) shows the Volume of Revolution tutor computing the volume of the solid by shells. The figure of the solid is correct, as is the computed volume. Note the selection of the vertical axis of rotation and its displacement of 2 units from the coordinate axis, and frame and scaling options applied in the Plot Options panel.
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The computation of the volume by the method of disks must be done from first principles.
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Figure 5.2.4(i) Volume of Revolution tutor
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Volume by the method of disks:
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Expression palette: Definite-integral template
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Context Panel: Evaluate and Display Inline
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=
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