Explanation Given: The functions f ( x ) = 1 + x , and g ( x ) = 2 − x , and the interval [ 0 , 3 ] . Concept Used: The exact area between the graphs of f andg on the interval [ a , b ] , where f > g , is given by the definite integral ∫ a b [ f ( x ) − g ( x ) ] d x . The area that is bounded by the graph of f ( x ) above the x-axis is positive. The area that is bounded by the graph of f ( x ) below the x-axis is negative. Power rule of integration: ∫ x n d x = x n + 1 n + 1 Calculation: In order tofind the exact area of the region between the graphs of f ( x ) = 1 + x , and g ( x ) = 2 − x on the interval [ 0 , 3 ] , first plot the graph of the functions and check the bounded region. Regions can be divided into subintervals as per the upper curve and lower curve. The area between any two graphs on any interval is given by the definite integral on that interval by subtracting the lower curve from upper curve. Graph of the functions f ( x ) = 1 + x , and g ( x ) = 2 − x on the interval [ 0 , 3 ] is shown below:

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 4.6, Problem 41S

To determine

The exact area of the region between the graphs of f(x)=1+x, and g(x)=2−x on the interval [0, 3] using definite integrals and the Fundamental Theorem of Calculus.

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Chapter 4.6, Problem 40S

Chapter 4.6, Problem 42S

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The exact area of the region between the graphs of f ( x ) = 1 + x , and g ( x ) = 2 − x on the interval [ 0 , 3 ] using definite integrals and the Fundamental Theorem of Calculus.

Solution Summary: The author explains the definite integrals and Fundamental Theorem of Calculus to determine the exact area of the region between the graphs of f(x)=1+x,

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The exact area of the region between the graphs of f(x)=1+x, and g(x)=2−x on the interval [0, 3] using definite integrals and the Fundamental Theorem of Calculus.

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