   Chapter 14.4, Problem 49E

Chapter
Section
Textbook Problem

Proof Prove the following Theorem of Pappus: Let R be a region in a plane and let L be a line in the same plane such that L does not intersect the interior of R. If r is the distance between the centroid of R and the line, then the volume V of the solid of revolution formed by revolving R about the line is V = 2 π r A , where A is the area of R.

To determine

To Prove: The Second Theorem of Pappus: Assume R to be a region in a plane and L to be a line in the same plane in such a manner that L does not intersect the interior of R. If r is the distance from the centroid of R to the line, then the volume V of the solid of revolution formed by revolving R about the line is written as V=2πrA, where A represents the area of R.

Explanation

Given:

R- The region of the given plane

L - A line in the same plane such that L does not intersect the interior of R

r- The distance of the centroid of R to the line

A - The area of R.

Proof:

Using the data given in the question, draw the figure with reference to xy-coordinate with the orientations that are shown below.

The volume of this solid can be written as:

V=R2πxdA

Now, solve the double integral to get:

V=R2πxdAV=2πRxdA

Now, multiply and divide V by RdA to get;

V=2

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