Explanation Given information: The provided curve is: f ( x ) = x 3 and g ( x ) = 2 x − x 2 . Formula used: Area between two curve: If f and g are continuous functions on [ a , b ] and if f ( x ) ≥ g ( x ) on [ a , b ] , then the area of the region bounded by y = f ( x ) , y = g ( x ) , x = a , and x = b is: A = ∫ a b [ f ( x ) − g ( x ) d x ] Or A = Integral of ( Top − Bottom ) Calculation: Consider the provided equation: f ( x ) = x 3 and g ( x ) = 2 x − x 2 Draw the graph of the above function to check whether f ( x ) ≥ g ( x ) : Calculate the values at different values of x : Substitute x = 0 in the function f ( x ) = x 3 . y = ( 0 ) 3 = 0 Substitute x = 1 in the function f ( x ) = x 3 . y = ( 1 ) 3 = 1 Substitute x = 2 in the function f ( x ) = x 3 . y = ( 2 ) 3 = 8 Substitute x = − 1 in the function f ( x ) = x 3 . y = ( − 1 ) 3 = − 1 The table provided below shows the values of the function at different values of x . x f ( x ) = x 3 Coordinates ( x , y ) 0 0 ( 0 , 0 ) 1 1 ( 1 , 1 ) 2 8 ( 2 , 8 ) − 1 − 1 ( − 1 , − 1 ) Now, consider g ( x ) = 2 x − x 2 Substitute x = 0 in the function g ( x ) = 2 x − x 2 . y = 2 ( 0 ) − ( 0 ) 2 = 0 Substitute x = 1 in the function g ( x ) = 2 x − x 2 . y = 2 ( 1 ) − ( 1 ) 2 = 1 Substitute x = 2 in the function g ( x ) = 2 x − x 2 . y = 2 ( 2 ) − ( 2 ) 2 = 0 Substitute x = − 1 in the function g ( x ) = 2 x − x 2 . y = 2 ( − 1 ) − ( − 1 ) 2 = − 3 The table provided below shows the values of the function at different values of x . x g ( x ) = 2 x − x 2 Coordinates ( x , y ) 0 0 ( 0 , 0 ) 1 1 ( 1 , 1 ) 2 0 ( 2 , 0 ) − 1 − 3 ( − 1 , − 3 ) Use the above tables and information to draw the graph of f ( x ) = x 3 and g ( x ) = 2 x − x 2 in the same graph so as to check whether f ( x ) ≥ g ( x ) :

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ISBN: 9780840058225

Author: HARSHBARGER

Publisher: CENGAGE L

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Chapter 13.3, Problem 22E

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To calculate: The area of the region between f(x)=x3 and g(x)=2x−x2.

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Chapter 13.3, Problem 21E

Chapter 13.3, Problem 23E

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Ch. 13.1 - CHECKPOINT 1. For the interval [0, 2]. determine...Ch. 13.1 - Prob. 2CP Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E

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The area of the region between f ( x ) = x 3 and g ( x ) = 2 x − x 2 .

Solution Summary: The author calculates the area of the region between the curve f(x)=x

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To calculate: The area of the region between f(x)=x3 and g(x)=2x−x2.

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Chapter 13 Solutions