Explanation 1) Concept: i. If the cross section is a washer with the inner radius r i n and the outer radius r o u t , then the area of the washer is obtained by subtracting the area of the inner disk from the area of the outer disk. A = π o u t e r r a d i u s 2 - π i n n e r r a d i u s 2 ii. The volume of a solid of revolution about the x -axis is V = ∫ a b A ( x ) d x 2) Given: y = 6 - x 2 , y = 2 ; about the x - axis 3) Calculation: The region bounded by y = 6 - x 2 a n d y = 2 and solid obtained by rotation about the x - axis are shown below. Here, the region is rotated about the x – axis, so the cross-section is perpendicular to the x -axis. A cross section of the solid is the washer with inner radius 2 and outer radius 6 - x 2 . So, its cross sectional area becomes A x = π o u t e r r a d i u s 2 - π i n n e r r a d i u s 2 = π 6 - x 2 2 - π 2 2 A x = π 6 - x 2 2 - 4 = π 36 - 12 x 2 + x 4 - 4 A x = π x 4 - 12 x 2 + 32 The region of integration is bounded by y = 6 - x 2 a n d y = 2 , At point of intersection of the curve and x-axis, 6 - x 2 = 2 x 2 = 4 x = ± 2 The solid lies between x = - 2 a n d

8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Chapter 5.2, Problem 8E

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The volume of the solid obtained by rotating the region bounded by the given curves about the x axis and sketch the region, the solid, and a typical disk or washer.

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Chapter 5.2, Problem 7E

Chapter 5.2, Problem 9E

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Chapter 5 Solutions

Calculus (MindTap Course List)

Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Prob. 3E Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 10E

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The volume of the solid obtained by rotating the region bounded by the given curves about the x axis and sketch the region, the solid, and a typical disk or washer.

Solution Summary: The author calculates the volume of the solid obtained by rotating the region bounded by the given curves about the x axis.

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To find: The volume of the solid obtained by rotating the region bounded by the given curves about the x axis and sketch the region, the solid, and a typical disk or washer.

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Chapter 5 Solutions

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the x axis and sketch the region, the solid, and a typical disk or washer.