Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
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 y=−1.
Figures 5.2.2(a-c) illustrate the essential steps in the method of disks as applied to this example. In Figure 5.2.2(a) the region A is shaded, with the arrows representing the radii of rotation. The black arrow corresponds to the outer radius R=1+x2; the green arrow, to the inner radius r=1.
Figure 5.2.2(a) Region A
Student:-Calculus1:-VolumeOfRevolution(x^2,0,0..1,axis=horizontal, distancefromaxis=-1,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=[-150,85,-10]);
Figure 5.2.2(b) The solid
Student:-Calculus1:-VolumeOfRevolution(x^2,0,0..1,axis=horizontal, distancefromaxis=-1,showvolume=false,showsum=true,showregion=false, method =midpoint,partition=6,output=plot,axes=frame,sumvolumeoptions=[color= brown,transparency=0,lightmodel=light3],caption="",tickmarks=[2,,5],labels=[x,z,y],scaling=constrained,orientation=[-150,85,-10]);
Figure 5.2.2(c) Disks
The solid of rotation itself is shown in Figure 5.2.2(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 5.2.2(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=1; the outer, R=1+x2.
One washer has volume π R2−r2 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+2 x2 ⅆx = 13⁢π15
Figure 5.2.2(d) shows the Volume of Revolution tutor applied to the given solid.
Note the inclusion of the bounding function gx=0, without which the volume would be incorrectly computed as 28 π/15.
The Plot Options button has been used to change the axes style (frame) and to set Constrained Scaling.
Because Maple can't determine which of R or r is greater, the absolute value of the difference R2−r2 is integrated.
Using the Calculus palette's definite-integral template, the volume of the solid of revolution (computed by the methods of disks) is
π ∫01x4+2 x2 ⅆx = 13⁢π15
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