Chapter 5: Applications of Integration
Section 5.2: Volume of a Solid of Revolution
Example 5.2.5
If A is the plane region bounded by the x-axis and the graphs of y=x2 and x=1, use the method of shells to calculate the volume of the solid of revolution formed when A is rotated about the x-axis.
Solution
Mathematical Solution
The animation in Figure 5.2.5(a) shows a single shell sweeping through the solid of revolution formed when region A is rotated about the x-axis.
The length (height) of the generic cylindrical shell for this solid is Ly=1−xy=1−y. This shell is formed from a rectangular slab dy thick, and of length Ly and width 2 π ρ, where ρ=y.
Thus, the volume of the slab from which the shell is constructed is 2 π ρ L dy=2 π y 1−y.
The volume of the solid of revolution is then 2 π ∫01y 1−y ⅆy = π5
The solid itself is shown in Figure 5.2.5(b).
use plots, plottools in module()
local q,F,Q,QQ,a1,a2,S;
q := plot(x^2, x = 0 .. 1, filled = true):
F := transform(proc (x, y) options operator, arrow; [x, y, 0] end proc):
Q := display(F(q)):
QQ := rotate(Q, Pi/2,[[0,0,0],[1,0,0]]):
a1 := animate(plot3d,[[x^2,t,z],t=0..2*Pi,z=x..1,coords=cylindrical,style=surface],x=0..0.98,paraminfo=false):
a2 := rotate(a1,Pi/2,[[0,0,0],[0,1,0]]):
S := display([QQ,a2],scaling=constrained,axes=frame,tickmarks=[2,[0],2],labels=[x,z,y],view=[0..1,-1..1,-1..1], orientation=[-125,65]);
print(S);
end module:
end use:
Figure 5.2.5(a) Animation of shells
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 5.2.5(b) The solid of revolution
Maple Solution
For rotation about a horizontal axis, the tutor provides only the method of disks.
Nevertheless, Figure 5.2.5(c) shows the Volume of Revolution tutor computing the volume of the solid by disks. The figure of the solid is correct, as is the computed volume. Note the selection of the horizontal axis of rotation, and frame and scaling options applied in the Plot Options panel.
The computation of the volume by the method of shells must be done from first principles.
Figure 5.2.5(c) Volume of Revolution tutor
Volume by the method of shells:
Expression palette: Definite-integral template
Context Panel: Evaluate and Display Inline
2 π ∫01y 1−y ⅆy = π5
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