Chapter 7: Additional Applications of Integration
Section 7.3: The Theorems of Pappus
Essentials
Table 7.3.1 lists the first and second theorems of Pappus, the first concerning the volume of a solid of revolution; the second, the surface area of a surface of revolution.
Theorem 1
R, a plane region of area A, is rotated about line L that does not intersect the interior of R
V is the volume of the solid of revolution so formed
P is the centroid of R
d is the distance that P traverses as R rotates about L
⇒
V=A⋅d
Theorem 2
C, a plane curve of length s, is rotated about line L that does not intersect C
S is the surface area of the surface of revolution so formed
P is the centroid of C
d is the distance that P traverses as C rotates about L
S=s⋅d
Table 7.3.1 The two theorems of Pappus
In effect, the first theorem of Pappus states that, under suitable conditions, the volume of a solid of revolution is equal to the product of the area of the region rotated to produce the solid, and the distance traversed by the centroid of this region.
Similarly, the second theorem of Pappus states that, under suitable conditions, the surface area of a surface of revolution is equal to the product of the length of the curve rotated to produce the surface, and the distance traversed by the centroid of the curve.
The centroid of a plane region and the centroid of a plane curve are detailed in Section 5.7.
The astute reader will realize that the two theorems of Pappus generally require two integrations, one to find the location of the centroid, and one to find the area or arc length. However, when symmetry and/or partitioning can be used to eliminate or simplify at least one integral, then applying a theorem of Pappus could be less work than using one of the direct methods in Section 5.2 or Section 5.5, which require only one integration.
Examples
Example 7.3.1
Use the first theorem of Pappus to find the volume of the solid of revolution formed when the plane region bounded by the x-axis and the graphs of y=x2 and x=1, is rotated about the x-axis. (See Example 5.2.1.)
Example 7.3.2
Use the second theorem of Pappus to find the surface area of the surface of revolution formed when the curve defined by y=x2,x∈0,1, is rotated about the x-axis. (See Example 5.5.1.)
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.
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