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
<< Previous Example Section 7.3
Next Chapter >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. 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
Download Help Document
What kind of issue would you like to report? (Optional)