Explanation Given: The equation of the curves are y = e x and y = 2 − x 2 . Calculation: Consider the given equations as follows. y = e x (1) y = 2 − x 2 (2) Plot a graph for the equation of the curves y = e x and y = 2 − x 2 as follows. Calculate y value using equation (1) Substitute 0 for x in equation (1). y = e 0 = 1 Hence, the co-ordinate of ( x , y ) is ( 0 , 1 ) . Substitute 1 for x in equation (1). y = e 1 = 2.718 Hence, the co-ordinate of ( x , y ) is ( 1 , 2.718 ) . Substitute 0 for x in Equation (2). y = 2 − ( 0 ) 2 = 2 The co-ordinate of ( x , y ) is ( 0 , 2 ) . Substitute 1 for x in Equation (2). y = 2 − ( 1 ) 2 = 1 The co-ordinate of ( x , y ) is ( 1 , 1 ) . Similarly calculate the coordinate values up to the bounded the region in the graph. By the use of the above points the region is drawn as shown below in Figure 1. From the Figure 1 the x -coordinates are ( − 1.316 , 0.537 ) . Hence, the approximate x -coordinates of the points of intersection of the curves ( a , b ) is ( − 1.316 , 0.537 ) _ . Calculate the area of the region: A = ∫ a b [ f ( x ) − g ( x ) ] d x (3) Substitute − 1.316 for a , 0.537 for b , ( 2 − x 2 ) for [ f ( x ) ] , and e x for [ g ( x ) ] in equation (3). A = ∫ − 1.316. 0.537 ( 2 − x 2 − e x ) d x (4) Integrate the above equation (4). A = [ 2 x − x 3 3 − e x ] − 1.316 0.537 = [ 2 ( 0.537 ) − ( 0.537 ) 3 3 − e 0.537 ] − [ 2 ( − 1.316 ) − ( − 1.316 ) 3 3 − e ( − 1.316 ) ] = ( − 0.688 + 2.140 ) = 1.452 Thus, the area A = 1.452 . Calculate the ( x ¯ ) coordinate of centroid: x ¯ = 1 A ∫ a b x [ f ( x ) − g ( x ) ] d x (5) Substitute − 1.316 for a , 0.537 for b , 1.452 for A , ( 2 − x 2 ) for [ f ( x ) ] , and e x for [ g ( x ) ] in equation (5). x ¯ = 1 1.452 ∫ − 1.316. 0.537 x ( 2 − x 2 − e x ) d x = 1 1.452 ∫ − 1.316. 0.537 ( 2 x − x 3 − x e x ) d x (6) Integrate equation (6). x ¯ = 1 1.452 [ 2 x 2 2 − x 4 4 − x e x + e x ] − 1

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Chapter 8.3, Problem 38E

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To find: The approximate x-coordinates of the points of intersection of the given curves using the graph and to find the approximate the centroid of the region bounded of the curves.

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Chapter 8.3, Problem 37E

Chapter 8.3, Problem 39E

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The approximate x -coordinates of the points of intersection of the given curves using the graph and to find the approximate the centroid of the region bounded of the curves.

Solution Summary: The author calculates the approximate x -coordinates of the points of intersection of a given curve using the graph.

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To find: The approximate x-coordinates of the points of intersection of the given curves using the graph and to find the approximate the centroid of the region bounded of the curves.

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Chapter 8 Solutions