Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
Example 5.2.1
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 x-axis.
Solution
Mathematical Solution
Figures 5.2.1(a-c) illustrate the essential steps in the method of disks. In Figure 5.2.1 the region A is shaded, and the arrow represents ρx, the varying radius of rotation. Here, this radius of rotation is given by the height of the curve y=x2.
module() local p1,p2,V,p3,p4; p1:=plot(x^2,x=0..1,filled=[color=brown,transparency=.4],color=black,labels=[x,y],tickmarks=[2,2],thickness=3): V:=VectorCalculus:-RootedVector(root=[3/4,0],<0,9/16>): p2:=VectorCalculus:-PlotVector(V,color=black): p3:=plots:-textplot([.88,.3,typeset(rho=x^2)],font=[default,12]): p4:=plots:-display(p2,p1,p3); print(p4); end module:
Figure 5.2.1(a) Region A
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]);
Figure 5.2.1(b) The solid
Student:-Calculus1:-VolumeOfRevolution(x^2,0..1,axis=horizontal, 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=[2,2,2],labels=[x,z,y],scaling=constrained);
Figure 5.2.1(c) Disks
The solid of rotation is shown in Figure 5.2.1(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 5.2.1(c) shows the solid sliced into a stack of disks. One disk has volume π ρ2 dx, leading to the definite integral listed in Table 5.2.1.
The actual volume, computed as per Table 5.2.1, is π ∫01x4 ⅆx = π5
Maple Solution
Figure 5.2.1(d) shows the tutor applied to the given solid. The Plot Options button has been used to change the axes style (frame) and to set Constrained Scaling.
In addition to the view of the solid shown in Figure 5.2.1(d), the other selections in the Display section will produce images of the plane region A being rotated, and of the disks generated by the particular partitioning chosen.
The definite integral whose value is the volume of the solid is displayed and evaluated. If the Disks option is chosen, this display is expanded to include the corresponding Riemann sum, as per Table 5.2.1(a).
Figure 5.2.1(d) Volume of Revolution tutor
∫01πx4ⅆx=15π=.6283185308
Approximatingsum:16π∑i=1616i-1124=.6138448223
Table 5.2.1(a) Volume and Riemann sum approximation
Note the VolumeOfRevolution command at the bottom of the tutor. This command will return the various graphs shown in the tutor, the inert integral whose value is the volume, or the value of this integral. This command, and its many options, was used to generate Figures 5.2.1(b) and 5.2.1(c).
Using the Calculus palette's definite-integral template, the volume of the solid of revolution (computed by the methods of disks) is
π ∫01x4 ⅆx = π5
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