Explanation Given: The equations are x = ( y − 1 ) 2 , x − y = 1 ; about x = − 1 . Calculation: Show the equations as below: x = ( y − 1 ) 2 (1) x − y = 1 (2) Plot a graph for the equations x = ( y − 1 ) 2 and x = y 2 + 1 using the calculation as follows: Calculate y value using Equation (1) Substitute 0 for x in Equation (1). 0 = ( y − 1 ) 2 0 = y − 1 y = 1 Hence, the co-ordinate of ( x , y ) is ( 0 , 1 ) . Calculate y value using Equation (1) Substitute 1 for x in Equation (1). 1 = ( y − 1 ) 2 1 = y − 1 ± 1 = y − 1 y = 1 ± 1 y 1 = 0 y 2 = 2 Hence, the co-ordinate of ( x , y ) is ( 1 , 0 ) and ( 1 , 2 ) . Calculate y value using Equation (2). Substitute 0 for x in Equation (2). 0 − y = 1 y = − 1 The co-ordinate of ( x , y ) is ( 0 , − 1 ) . Calculate y value using Equation (2). Substitute 1 for x in Equation (2). 1 − y = 1 y = 1 − 1 = 0 The co-ordinate of ( x , y ) is ( 1 , 0 ) . Similarly calculate the coordinate values up to bound the region in the graph. Draw the washer as shown in Figure 1. Calculate the volume using the method of washer: V = ∫ a b π { [ f ( y ) ] 2 − [ g ( y ) ] 2 } d y (3) Calculate the y value: Substitute ( y − 1 ) 2 for x in Equation (2). ( y − 1 ) 2 − y = 1 y 2 − 2 y + 1 − y − 1 = 0 y 2 − 3 y = 0 y ( y − 3 ) = 0 (4) Solve Equation (4). y 1 = 0 y 2 = 3 Consider g ( y ) = − x + ( y − 1 ) 2 (5) Substitute ( − 1 ) for x in Equation (5). g ( y ) = − ( − 1 ) + ( y − 1 ) 2 = ( y − 1 ) 2 + 1 = y 2 − 2 y + 1 + 1 = y 2 − 2 y + 2 Consider f ( y ) = − x + y + 1 (6) Substitute ( − 1 ) for x in Equation (6)

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Chapter 6.3, Problem 43E

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To find: The volume of the resulting solid by the region bounded by given curve using the method of washer.

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Chapter 6.3, Problem 42E

Chapter 6.3, Problem 44E

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Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals, Volume I

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....

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The volume of the resulting solid by the region bounded by given curve using the method of washer.

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To find: The volume of the resulting solid by the region bounded by given curve using the method of washer.

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Chapter 6 Solutions