Chapter 7: Additional Applications of Integration
Section 7.3: The Theorems of Pappus
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.
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
Figure 7.3.3(a) Circular disk C and the axis of rotation
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