|
Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
|
Example 5.2.2
|
|
|
If is the plane region bounded by the -axis and the graphs of and , use the method of disks to calculate the volume of the solid of revolution formed when is rotated about the line .
|
|
|
|
|
Solution
|
|
|
|
Mathematical Solution
|
|
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 is shaded, with the arrows representing the radii of rotation. The black arrow corresponds to the outer radius ; the green arrow, to the inner radius .
|
>
|
module()
local p1,p2,VR,Vr,p3,p4,p5;
p1:=plot(x^2,x=0..1,filled=[color=brown,transparency=.4],color=black,labels=[x,y],tickmarks=[2,3],thickness=3):
p2:=plot(-1,x=0..1,color=black,thickness=2):
VR:=VectorCalculus:-RootedVector(root=[3/4,-1],<0,25/16>):
Vr:=VectorCalculus:-RootedVector(root=[1/4,-1],<0,1>):
p3:=VectorCalculus:-PlotVector([VR,Vr],color=[black,green],width=.03):
p4:=plots:-textplot({[.95,-.3,typeset(R=1+x^2)],[.36,-.3,typeset(r=1)]},font=[default,12]):
p5:=plots:-display(p3,p1,p2,p4);
print(p5);
end module:
|
|
|
Figure 5.2.2(a) Region
|
|
|
|
|
>
|
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,[0],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 is drawn on the surface of the solid. Note how the -axis is out of the -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 ; the outer, .
One washer has volume , leading to the definite integral listed in Table 5.2.1.
The actual volume, computed as per Table 5.2.1, is =
|
|
|
Maple Solution
|
|
|
•
|
Figure 5.2.2(d) shows the Volume of Revolution tutor applied to the given solid.
|
|
•
|
Note the inclusion of the bounding function , without which the volume would be incorrectly computed as .
|
|
•
|
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 or is greater, the absolute value of the difference is integrated.
|
|
•
|
Using the Calculus palette's definite-integral template, the volume of the solid of revolution (computed by the methods of disks) is
|
=
|
|
|
|
Figure 5.2.2(d)
tutor
|
|
|
|
|
|
|
|
|
<< Previous Example Section 5.2
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
|
