Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
Example 5.2.4
If A is the plane region bounded by the x-axis and the graphs of y=x2 and x=1, use the method of disks to calculate the volume of the solid of revolution formed when A is rotated about the line x=2.
Solution
Mathematical Solution
Figure 5.2.4(a) shows the region A, the axis of rotation, and the inner and outer radii r=1 and R=xy=2−y, respectively.
use plots,VectorCalculus in
module()
local q1,q2,q3,q4,q5,q6,q7,q8,q9;
q1 := plot(x^2,x=0..1,filled=true,color=red, view=[0..4,0..1],transparency=.5, tickmarks=[3,2], labels=[x,y]);
q2 := RootedVector(root=[2,0],<0,1>);
q3 := PlotVector(q2,color=black);
q4 := RootedVector(root=[2,.2],<-1.51,0>):
q5 := PlotVector(q4,color=blue):
q6 := RootedVector(root=[2,.7],<-1,0>):
q7 := PlotVector(q6,color=green):
q8 := textplot({[1.5,.37,typeset(R)],[1.5,.86,typeset(r)]}):
q9 := display([q1,q3,q5,q7,q8],scaling=constrained):
print(q9);
end module:
end use:
Figure 5.2.4(a) Region A
use plots, plottools,VectorCalculus in
local q,F,Q,A,p,V,k,spin;
q := plot(x^2,x=0..1,filled=true);
F := transform(proc (x,y) options operator, arrow; [x,0,y] end proc);
Q:=display(F(q));
V:=RootedVector(root=[2,0,0],[0,0,1]):
A:=PlotVector(V,color=black,width=.1):
p:=display([seq(rotate(Q,(2*Pi*(1/30))*k,[[2,0,0],[2,0,1]]),k=0..29)], insequence = true);
spin := display([A, p], labels = [x,z,y], axes = frame, scaling = constrained,orientation=[-70,72], tickmarks=[3,[0],2]);
print(spin):
Figure 5.2.4(b) Region A rotated
use VectorCalculus,plots, plottools in
local q,F,Q,V,A,p,spin,p4,p5,k;
q:=plot(x^2,x=0..1,filled=[color=red],color=red):
F:=transform((x,y) ->[x,0,y]):
Q:=display(F(q)):
p:=display([seq(rotate(Q,(2*Pi*(1/30))*k,[[2,0,0],[2,0,1]]),k=0..10)], insequence=true):
spin:=display([A, p]):
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):
p5:=display([p4,spin],labels=[x,z,y],tickmarks=[4,[0],2], orientation=[-55,70], axes=frame,scaling=constrained);
print(p5);
Figure 5.2.4(c) Solid generated
Figures 5.2.4(b) and 5.2.4(c) are animations, the first showing the region A rotated about the axis x=2; the second, the solid evolving from the rotation of region A about the line x=2.
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 y=x2 about the line x=2 (Figure 5.2.4(d)), the cylinder of radius 1 generated by rotating x=1 about the same axis (Figure 5.2.4(e)).
use plots, plottools in
local p1,F,G,p2,p3,p4,p5,p6;
p1 := disk([0,0],1,color=green): p6 := disk([0,0],2,color=green):
F := transform((x,y)->[x,y,1]): G := transform((x,y)->[x,y,-.02]):
p2 := plot3d([r,t,(r-2)^2],r=1..2,t=0..2*Pi,coords=cylindrical,style=surface, color=red):
p3 := display([F(p1),p2,G(p6)]);
p4 := translate(p3,2,0,0);
p5 := display(p4, axes=frame, tickmarks=[3,[0],[0,1]], labels=[x,z,y], orientation=[-50,70], scaling=constrained);
Figure 5.2.4(d) Rotate y=x2
local p3,p4,p5,p6,f,g;
p3 := plot3d([1,t,z],t=0..2*Pi,z=0..1,coords=cylindrical, style=surface, color=gold);
p5 := disk([2,0],1,color=green);
f := transform((x,y)->[x,y,1]); g := transform((x,y)->[x,y,0]):
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]);
print(p6);
Figure 5.2.4(e) Rotate x=1
use plots, plottools, Student[Calculus1] in
local p6, p7, p8, p9;
p6 := VolumeOfRevolution(x^2,0, 0..1, 'axis'='vertical', 'distancefromaxis' = 2, 'output'='plot',title="",volumeoptions=[color=red,transparency=0],caption="",lineoptions=[color=black]):
p7 := plot3d([.99,t,z],t=0..2*Pi,z=0..1,coords=cylindrical,style=surface,color=gold):
p8 := display([p6,translate(p7,2,0,0)]);
p9 := display(p8, view=0..1,scaling=constrained,axes=frame, labels=[x,z,y],orientation=[-125,70,0],tickmarks=[3,[0],2]);
print(p9);
Figure 5.2.4(f) Composite solid
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) dy, and inner and outer radii r=1 and R=2− y, respectively.
use Student[Calculus1], plots, plottools in
local p1,p2;
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]):
p2 := display(rotate(p1,0,Pi/2,0),orientation=[-90,50]);
print(p2);
Figure 5.2.4(g) Stack of washers
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]):
p2 := display(rotate(p1,0,Pi/2,0),orientation=[-90,55]);
Figure 5.2.4(h) Single washer
As per Table 5.2.1, the volume of the solid is obtained by the following calculation.
V
=π ∫01R2y ⅆy−π ∫01r2y ⅆy
=π ∫01R2y−r2y ⅆy
=π ∫012−y2−1 ⅆy
=56 π
Maple Solution
For rotation about a vertical axis, the tutor provides only the method of shells.
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.
The computation of the volume by the method of disks must be done from first principles.
Figure 5.2.4(i) Volume of Revolution tutor
Volume by the method of disks:
Expression palette: Definite-integral template
Context Panel: Evaluate and Display Inline
π ∫012−y2−1 ⅆy = 56π
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