Chapter 7: Additional Applications of Integration
Section 7.3: The Theorems of Pappus
Example 7.3.3
If R>r, and C is a circular disk of radius r, rotating C about a line that is in the plane of C and at a distance R from the center of C, forms a torus. Use the first theorem of Pappus to find the volume of this torus.
Solution
Figure 7.3.3(a) shows the circular disk C, the axis of rotation, and the relative lengths R and r.
By symmetry, the centroid of the disk is the center of the circle, which will trace a circle of radius of R, and hence traverse a distance of 2 π R as the torus is formed.
The area of the disk is π r2, so, by the first theorem of Pappus, the volume of the torus is 2 π R⋅π r2=2 π2r2R.
Note that here, a theorem of Pappus completely eliminated the need for integration!
use plots, plottools, Student[VectorCalculus] in module() local p1,p2,p3,p4,p5,V1,V2; p1:=circle([4,0],1): p2:=line([1,-2],[1,2],thickness=2,color=black): p3:=plot([[4,0]],style=point,symbol=solidcircle,symbolsize=20,color=black): V1:=RootedVector(root=[1,0],<3,0>): V2:=RootedVector(root=[4,0],<-1,1>/sqrt(2)): p4:=PlotVector([V1,V2],color=[red,green],width=.1): p5:=textplot({[2,.3,typeset(R)],[3.8,.5,typeset(r)],[4,1.2,typeset(C)]},font=[default,14]): print(display(p1,p2,p3,p4,p5,scaling=constrained,axes=none)); end module: end use:
Figure 7.3.3(a) Circular disk C and the axis of rotation
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