Explanation Show the ring is divided into three segments as in Figure 1. Express the surface area of the ring using the theorem of Pappus and Guldinus. A = θ × ∑ r ¯ L (I) Here, the surface area of revolution is A , the angle of revolution measured in radians is θ , the perpendicular distance from the axis of revolution to the centroid of the generating curve is r ¯ , and the length of the generating curve is L . Since the angle of revolution is measured in radians, θ = 2 π rad Substitute 2 π rad for θ in Equation (I). A = 2 π × ∑ r ¯ L (II) Conclusion: Refer to Figure 1 and use spreadsheet to calculate the values of L , r ¯ , and r ¯ L for three segments as shown in Table 1. Segment L ( mm ) r ¯ ( mm ) r ¯ L ( mm 2 ) 1

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Chapter 9.3, Problem 99P

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The surface area of the ring (A) .

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Chapter 9.3, Problem 98P

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The surface area of the ring ( A ) .

Solution Summary: The author explains the surface area of the ring using the theorem of Pappus and Guldinus.

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