Explanation Definition used: Definition of Volume by cylindrical shell method: Consider that the solid S is obtained by rotating the region under the curve y = f ( x ) about the y -axis from a to b is V = ∫ a b π A ( y ) d y and about the x -axis from a to b is V = ∫ a b π A ( x ) d x . Calculation: The figure has a bigger rectangle of length d and width is b. When this rectangle is rotated about x -axis, the volume of the bigger cylinder is computed as V B = π b 2 d . The figure has a smaller rectangle of length c and width is a. When this rectangle is rotated about x -axis, the volume of the smaller cylinder is computed as V S = π a 2 c . Now, the region bounded between y = c , y = d and x = f − 1 ( y ) can be obtained as follows: Compute the area of the disk with radius x = f − 1 ( y ) as π [ f − 1 ( y ) ] 2 . Integrate this area along y -axis from c to d and obtain the volume of the solid obtained by rotating curve about y -axis as V S o l i d = ∫ c d π [ f − 1 ( y ) ] 2 d y . Thus, the volume of the shaded region V becomes V = V B − V S − V S o l i d = π b 2 d − π a 2 c − ∫ c d π [ f − 1 ( y ) ] 2 d y Hence, it is proved that the V = π b 2 d − π a 2 c − ∫ c d π [ f − 1 ( y ) ] 2 d y . To prove: V = ∫ a b 2 π x f ( x ) d x . When y = f ( x ) is substituted, the volume of the solid is obtained by rotating curve about x -axis

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 7.3, Problem 43E

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To show: The volume V is πb2d−πa2c−∫cdπ[f−1(y)]2dy and also, prove V=∫ab2πxf(x)dx using integration by parts on the resultant integral.

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Chapter 7.3, Problem 42E

Chapter 7.4, Problem 1E

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Chapter 7 Solutions

Essential Calculus: Early Transcendentals

Ch. 7.1 - Find the area of the shaded region.Ch. 7.1 - Prob. 2E Ch. 7.1 - Find the area of the shaded region.Ch. 7.1 - Find the area of the shaded region.Ch. 7.1 - Prob. 5E Ch. 7.1 - Sketch the region enclosed by the given curves....Ch. 7.1 - Prob. 7E Ch. 7.1 - Sketch the region enclosed by the given curves....Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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The volume V is π b 2 d − π a 2 c − ∫ c d π [ f − 1 ( y ) ] 2 d y and also, prove V = ∫ a b 2 π x f ( x ) d x using integration by parts on the resultant integral.

Solution Summary: The author illustrates the volume by cylindrical shell method by rotating the region under the curve y=f(x).

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To show: The volume V is πb2d−πa2c−∫cdπ[f−1(y)]2dy and also, prove V=∫ab2πxf(x)dx using integration by parts on the resultant integral.

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Chapter 7 Solutions