Explanation Given: The equations of the curve are x + y = 2 and x = y 2 . Calculation: Consider the equations as below: x + y = 2 (1) x = y 2 (2) Plot a graph for the equations x + y = 2 and x = y 2 using the calculation as follows: Calculate y value using Equation (1) Substitute 0 for x in Equation (1). 0 + y = 2 y = 2 − 0 = 2 Hence, the co-ordinate of ( x , y ) is ( 0 , 2 ) . Substitute 1 for x in Equation (1). 1 + y = 2 y = 2 − 1 = 1 Hence, the co-ordinate of ( x , y ) is ( 1 , 1 ) . Substitute 0 for x in Equation (2). 0 = y 2 y = 0 The co-ordinate of ( x , y ) is ( 0 , 0 ) . Substitute 1 for x in Equation (2). 1 = y 2 y = 1 = ± 1 The co-ordinate of ( x , y ) is ( 1 , ± 1 ) . Draw the region as shown in Figure 1. Calculate the area of the region: A = ∫ a b [ f ( y ) − g ( y ) ] d y (3) Calculate the y values: Substitute y 2 for x in Equation (1). y 2 + y = 2 y 2 + y − 2 = 0 y 2 − y + 2 y − 2 = 0 y ( y − 1 ) + 2 ( y − 1 ) = 0 ( y − 2 ) + ( y − 1 ) = 0 y 1 = − 2 y 2 = 1 Rearrange Equation (1). x = 2 − y Substitute ( − 2 ) for a , 1 for b , ( 2 − y ) for [ f ( y ) ] , and y 2 for [ g ( y ) ] in Equation (3). A = ∫ − 2 1 ( 2 − y − y 2 ) d x (4) Integrate Equation (4). A = [ 2 y − y 1 + 1 1 + 1 − y 2 + 1 2 + 1 ] − 2 1 = [ 2 y − y 2 2 − y 3 3 ] − 2 1 = [ 2 ( 1 ) − ( 1 ) 2 2 − ( 1 ) 3 3 ] − [ 2 ( − 2 ) − ( − 2 ) 2 2 − ( − 2 ) 3 3 ] = [ 2 − 1 2 − 1 3 ] − [ − 4 − 2 + 8 3 ] = 9 2 Calculate the ( x ¯ ) coordinate of the centroid: x ¯ = 1 A ∫ a b 1 2 { [ f ( y ) ] 2 − [ g ( y ) ] 2 } d x (5) Substitute ( − 2 ) for a , 1 for b , 9 2 for A , ( 2 − y ) for [ f ( y ) ] , and y 2 for [ g ( y ) ] in Equation (5). x ¯ = 1 9 2 ∫ − 2 1 1 2 { ( 2 − y ) 2 − ( ( y 2 ) ) 2 } d y = 2 9 × 1 2 ∫ − 2 1 1 2 [ ( 4 − 4 y + y 2 ) − y 4 ] d y = 1 9 ∫ − 2 1 ( − y 4 + y 2 − 4 y + 4 ) d y (6) Integrate Equation (6)

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 8.3, Problem 33E

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To find: The centroid (x¯,y¯) of the region bounded by the given curve.

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Chapter 8.3, Problem 32E

Chapter 8.3, Problem 34E

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Chapter 8 Solutions

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The centroid ( x ¯ , y ¯ ) of the region bounded by the given curve.

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To find: The centroid (x¯,y¯) of the region bounded by the given curve.

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