Explanation Given: The function f ( x ) = sin x + x cos x , and the interval [ − π , π ] . Concept Used: The average value of an integrable functionf on the interval [ a , b ] , is given by 1 b − a ∫ 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: Observe that the average value of an integrable function f on the interval [ a , b ] , is given by 1 b − a ∫ a b f ( x ) d x , so using this concept the exact average value of f ( x ) = sin x + x cos x on the interval [ − π , π ] is given by, f avg = 1 π −

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 4.6, Problem 67S

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The exact average value of f(x)=sinx+xcosx on the interval [−π, π] using definite integrals and the Fundamental Theorem of Calculus.

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Chapter 4.6, Problem 66S

Chapter 4.6, Problem 68S

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The exact average value of f ( x ) = sin x + x cos x on the interval [ − π , π ] using definite integrals and the Fundamental Theorem of Calculus.

Solution Summary: The author calculates the average value of an integrable function f on the interval left[a,bright] using definite integrals and the Fundamental Theorem of Calculus.

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The exact average value of f(x)=sinx+xcosx on the interval [−π, π] using definite integrals and the Fundamental Theorem of Calculus.

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