Explanation An arc of a curve revolving about a line generates a surface called a surface of revolution. If the generating arc is a line segment, the frustum of a cone is generated. A cone with slant height s and the base readius r has lateral surface area A = π r s . cut the cone along the dashed line and flatten it out as shown in. The circumference of the circular base of the cone is 2 π r . Which is same as the arc of the length s θ after the cone is flattened. Thus, 2 π r = s θ AND θ = 2 π r s The lateral surface area of the cone is the ares of sector COD. That is , A = 1 2 s 2 θ = 1 2 s 2 ( 2 π r s ) The slant height of the frustum e cone is I, top radius is of r 1 and bottom radius r 2 . Remove a small cone of base radius r 1 and slant height l 1 from the larger cone of base radius r 2 and slant height l 2 = l 1 + l from the frustum of the cone. Find the lateral surface area F of the frustum by subtracting the area of the surface of larger cone. That is , F = π r 2 l 2 − π r 1 l 1 F = π r 2 ( l 1 + l ) − π r 1 l 1 F = π [ ( r 2 − r 1 ) l 1 + r 2 l ] Using similar triangles see that l 1 r 1 = l 1 + l r 2 Thus, F = π [ ( r 2 − r 1 ) l 1 + r 2 l ] F = π ( r 2 l + r 1 l ) F = 2 π ( r 1 + r 2 2 ) l Where 1 2 ( r 1 + r 2 ) is the average of the radii of the frustum

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6, Problem 10PE

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To find: The formula for area of a surface of revolution.

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Chapter 6, Problem 9PE

Chapter 6, Problem 11PE

Chapter 6 Solutions

Calculus

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The formula for area of a surface of revolution.

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Chapter 6 Solutions