The Second Theorem of Pappus is in the same spirit as Pappuss Theorem discussed in this section,

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The Second Theorem of Pappus is in the same spirit as Pappus’s Theorem discussed in this section, but for surface area rather than volume: let C be a curve that lies entirely on one side of a line l in the plane. If C is rotated about l, then the area of the resulting surface is the product of the arc length of C and the distance traveled by the centroid of C.

Use the Second Theorem of Pappus to find the surface area of the torus in Example 7.

EXAMPLE 7

A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R(> r) from the center of the circle. Find the volume of the torus.

The circle has area A = πr2. By the symmetry principle, its centroid is its center and so the distance traveled by the centroid during a rotation is d = 2πR. Therefore, by the Theorem of Pappus, the volume of the torus is

V = Ad = (2πR)(πr2) = 2π2r2R

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Calculus Early Transcendentals

ISBN: 9781337613927

9th Edition

Authors: James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

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