Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
Example 5.2.3
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 y-axis.
Solution
Mathematical Solution
Figures 5.2.3(a-c) illustrate the essential steps in the method of disks as applied to this example. In Figure 5.2.3(a) the region A is shaded, with the arrows representing the radii of rotation. The black arrow corresponds to the outer radius R=1; the green arrow, to the inner radius r=xy=y.
module() local p1,VR,Vr,p3,p4,p5; p1:=plot(x^2,x=0..1,filled=[color=brown,transparency=.4],color=black,labels=[x,y],tickmarks=[2,2],thickness=3): VR:=VectorCalculus:-RootedVector(root=[0,3/4],<1,0>): Vr:=VectorCalculus:-RootedVector(root=[0,1/4],<1/2,0>): p3:=VectorCalculus:-PlotVector([VR,Vr],color=[black,green],width=.03): p4:=plots:-textplot({[.25,.83,typeset(R=1)],[.25,.34,typeset(r=sqrt(y))]},font=[default,12]): p5:=plots:-display(p3,p1,p4); print(p5); end module:
Figure 5.2.3(a) Region A
Student:-Calculus1:-VolumeOfRevolution(x^2,0..1,axis=vertical, distancefromaxis=0,showvolume= true,showregion=true,output=plot,axes=frame,caption= "",volumeoptions=[color=red,transparency=0],scaling=constrained,tickmarks=[2,[-3,0,3],[-3,-2,-1,0,1]],labels=[x,z,y],orientation=[-105,45,0]);
Figure 5.2.3(b) The solid
Student:-Calculus1:-VolumeOfRevolution(x^2,0,0..1,axis=vertical, distancefromaxis=0,showvolume=false,showsum=true,showregion=false, method =midpoint,partition=6,output=plot,axes=frame,sumvolumeoptions=[color= brown,transparency=0,lightmodel=light3],caption="",tickmarks=[2,[0],2],labels=[x,z,y],scaling=constrained,orientation=[-105,45,0]);
Figure 5.2.3(c) Disks
The solid of rotation itself is shown in Figure 5.2.3(b). The bounding curve y=x2 is drawn on the surface of the solid. Note how the z-axis is out of the xy-plane, which is the plane of the viewing screen. Figure 5.2.3(c) shows the solid sliced into a stack of disks. Each such disk has a hole, so the punctured disk resembles a washer. The inner radius of the washer is r=xy=y; the outer, R=1.
One washer has volume π R2−r2 dy, leading to the definite integral listed in Table 5.2.1.
The actual volume, computed as per Table 5.2.1, is π ∫011−y ⅆy = π2
Maple Solution
For rotation about a vertical axis, the tutor provides only the method of shells.
Nevertheless, Figure 5.2.3(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 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 5.2.3(d) Volume of Revolution tutor
Volume by the method of disks:
Expression palette: Definite-integral template
Context Panel: Evaluate and Display Inline
π ∫011−y ⅆy = π2
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