Part (a): Explanation Given: The equations are y = cos 4 x , y = − cos 4 x , − π 2 ≤ x ≤ π 2 ; about x = π . Show the theorem 7 (Integrals of symmetric Functions): Apply the condition as follows if f is continuous on [ − a , a ] . a) If f is even function [ f ( − x ) = f ( x ) ] , then ∫ − a a f ( x ) d x = 2 ∫ 0 a f ( x ) d x . b) If f is odd function [ f ( − x ) = − f ( x ) ] , then ∫ − a a f ( x ) d x = 0 . Calculation: Show the equations as below: y = cos 4 x (1) y = − cos 4 x (2) Plot a graph for the equations y = cos 4 x and y = − cos 4 x using the calculation as follows: Calculate y value using Equation (1) Substitute ( − π 2 ) for x in Equation (1). y = cos 4 ( − π 2 ) = cos 4 ( π 2 ) = 0 Hence, the co-ordinate of ( x , y ) is ( − π 2 , 0 ) . Calculate y value using Equation (1) Substitute 0 for x in Equation (1). y = cos 4 ( 0 ) = 1 Hence, the co-ordinate of ( x , y ) is ( 0 , 1 ) . Calculate y value using Equation (1) Substitute ( π 2 ) for x in Equation (1). y = cos 4 ( π 2 ) = 0 Hence, the co-ordinate of ( x , y ) is ( π 2 , 0 ) . Calculate y value using Equation (2) Substitute ( − π 2 ) for x in Equation (2). y = − cos 4 ( − π 2 ) = − cos 4 ( π 2 ) = 0 Hence, the co-ordinate of ( x , y ) is ( − π 2 , 0 ) . Calculate y value using Equation (2) Substitute 0 for x in Equation (2). y = − cos 4 ( 0 ) = − 1 Hence, the co-ordinate of ( x , y ) is ( 0 , − 1 ) . Calculate y value using Equation (2) Substitute ( π 2 ) for x in Equation (2). y = − cos 4 ( π 2 ) = 0 Hence, the co-ordinate of ( x , y ) is ( π 2 , 0 ) . Draw the cylindrical shell as shown in Figure 1. Calculate the volume using the method of cylindrical shell: V = ∫ a b 2 π x [ f ( x ) ] d x (3) Consider x = π (4) Rearrange Equation (4) Part (b): To determine To evaluate: The integral function.

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ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 5.3, Problem 23E

Part (a):

To determine

To set up: an integral for the volume of the solid obtained by rotating the region bounded by the given curve.

Part (b):

To determine

To evaluate: The integral function.

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Chapter 5.3, Problem 22E

Chapter 5.3, Problem 24E

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To set up: an integral for the volume of the solid obtained by rotating the region bounded by the given curve.

Solution Summary: The author explains how to set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve.

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To set up: an integral for the volume of the solid obtained by rotating the region bounded by the given curve.

To evaluate: The integral function.

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Chapter 5 Solutions