a. Explanation Given: The function f ( x ) = 2 − e − x , and the interval [ − 1 , 0 ] . Concept Used: The signed area between the graph of f and the x -axis on the interval [ a , b ] is given by the definite integral ∫ a b f ( 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 value of the signed area of the region between the graph of f ( x ) = 2 − e − x and the x -axis on the interval [ − 1 , 0 ] , first plot the graph of the function f ( x ) = 2 − e − x on the interval [ − 1 , 0 ] . And check whether the area bounded by the graph is above the x -axis or below the x -axis. If the region bounded is above the x -axis, then the area will be positive and if it is below x -axis then the area would be negative. Graph of the function f ( x ) = 2 − e − x on the interval [ − 1 , 0 ] is shown below: From the above graph it is clear that the region A is above the x -axis and hence positive. And the region B is below the x -axis and hence negative. Since the signed area between the graph of f and the x -axis on the interval [ a , b ] is given by the definite integral ∫ a b f ( x ) d x , so using this concept the signed area between the graph of f ( x ) = 2 − e − x and the x -axis on the interval [ − 1 , 0 ] is given by ∫ − 1 0 ( 2 − e − x ) d x = ∫ − 1 ln ( 1 / 2 ) ( 2 − e − x ) d x + ∫ ln ( 1 / 2 ) 0 ( 2 − e − x ) d x = [ 2 x + e − x ]</ b. To determine The exact value of the absolute area of the region between the graph of f ( x ) = 2 − e − x and the x -axis on the interval [ − 1 , 0 ] using definite integrals and the Fundamental Theorem of Calculus.

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.6, Problem 34S

To determine

The exact value of the signed area of the region between the graph of f(x)=2−e−x and the x -axis on the interval [−1, 0] using definite integrals and the Fundamental Theorem of Calculus.

To determine

The exact value of the absolute area of the region between the graph of f(x)=2−e−x and the x -axis on the interval [−1, 0] using definite integrals and the Fundamental Theorem of Calculus.

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Chapter 4.6, Problem 33S

Chapter 4.6, Problem 35S

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The exact value of the signed area of the region between the graph of f ( x ) = 2 − e − x and the x -axis on the interval [ − 1 , 0 ] using definite integrals and the Fundamental Theorem of Calculus.

Solution Summary: The author calculates the exact value of the signed area between the graph of f(x)=2-e-x and the x -axis on the interval.

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The exact value of the signed area of the region between the graph of f(x)=2−e−x and the x -axis on the interval [−1, 0] using definite integrals and the Fundamental Theorem of Calculus.

The exact value of the absolute area of the region between the graph of f(x)=2−e−x and the x -axis on the interval [−1, 0] using definite integrals and the Fundamental Theorem of Calculus.

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