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

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ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 8.3, Problem 29E

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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 28E

Chapter 8.3, Problem 30E

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Single Variable Calculus

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

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