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Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
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Example 5.2.5
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If is the plane region bounded by the -axis and the graphs of and , use the method of shells to calculate the volume of the solid of revolution formed when is rotated about the -axis.
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Solution
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Mathematical Solution
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The animation in Figure 5.2.5(a) shows a single shell sweeping through the solid of revolution formed when region is rotated about the -axis.
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The length (height) of the generic cylindrical shell for this solid is . This shell is formed from a rectangular slab thick, and of length and width , where .
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Thus, the volume of the slab from which the shell is constructed is .
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The volume of the solid of revolution is then =
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The solid itself is shown in Figure 5.2.5(b).
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use plots, plottools in
module()
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local q,F,Q,QQ,a1,a2,S;
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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, y, 0] end proc):
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QQ := rotate(Q, Pi/2,[[0,0,0],[1,0,0]]):
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a1 := animate(plot3d,[[x^2,t,z],t=0..2*Pi,z=x..1,coords=cylindrical,style=surface],x=0..0.98,paraminfo=false):
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a2 := rotate(a1,Pi/2,[[0,0,0],[0,1,0]]):
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S := display([QQ,a2],scaling=constrained,axes=frame,tickmarks=[2,[0],2],labels=[x,z,y],view=[0..1,-1..1,-1..1], orientation=[-125,65]);
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Figure 5.2.5(a) Animation of shells
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Student:-Calculus1:-VolumeOfRevolution(x^2,0..1,axis=horizontal, distancefromaxis=0,showvolume= true,showregion=true,output=plot,axes=frame,caption= "",volumeoptions=[color=red,transparency=0],scaling=constrained,tickmarks=[2,[-3,0,3],3],labels=[x,z,y]);
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Figure 5.2.5(b) The solid of revolution
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Maple Solution
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For rotation about a horizontal axis, the
tutor provides only the method of disks.
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Nevertheless, Figure 5.2.5(c) shows the Volume of Revolution tutor computing the volume of the solid by disks. The figure of the solid is correct, as is the computed volume. Note the selection of the horizontal axis of rotation, and frame and scaling options applied in the Plot Options panel.
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The computation of the volume by the method of shells must be done from first principles.
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Figure 5.2.5(c) Volume of Revolution tutor
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Volume by the method of shells:
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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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