a. Explanation Given: The function f ( x ) = 1 1 + x 2 , and the interval [ − 1 , 1 ] . 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 ) = 1 1 + x 2 and the x -axis on the interval [ − 1 , 1 ] , first plot the graph of the function f ( x ) = 1 1 + x 2 on the interval [ − 1 , 1 ] . 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 ) = 1 1 + x 2 on the interval [ − 1 , 1 ] is shown below: b. To determine The exact value of the absolute area of the region between the graph of f ( x ) = 1 1 + x 2 and the x -axis on the interval [ − 1 , 1 ] using definite integrals and the Fundamental Theorem of Calculus.

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 4.6, Problem 37S

To determine

The exact value of the signed area of the region between the graph of f(x)=11+x2 and the x -axis on the interval [−1, 1] 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)=11+x2 and the x -axis on the interval [−1, 1] using definite integrals and the Fundamental Theorem of Calculus.

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Chapter 4.6, Problem 36S

Chapter 4.6, Problem 38S

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The exact value of the signed area of the region between the graph of f ( x ) = 1 1 + x 2 and the x -axis on the interval [ − 1 , 1 ] 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) 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)=11+x2 and the x -axis on the interval [−1, 1] 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)=11+x2 and the x -axis on the interval [−1, 1] using definite integrals and the Fundamental Theorem of Calculus.

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