a. Explanation Given Information: A function f is an integrable on [ a , b ] and x k = a + k ( b − a ) n . Proof: The accumulation function of the area A ( x ) can be written using the definition of the definite integral to find the area under the curve between the function and the x − axis on the interval [ a , b ] as given below. The exact value of the area on the interval [ a , b ] A ( x ) = ∫ a b f ( x ) d x The average of the area on the interval [ a , b ] = 1 b − a ∫ a b f ( x ) d x The approximation for the sum of areas of small rectangles with width 1 n and height f ( x n ) where n = 1 , 2 , 3 , ..... , n on the interval [ a , b ] can be calculated as given below. A ( x ) = 1 n . f ( x 1 ) + 1 n b. To determine To Explain: the left hand sideof the equation is a calculationof average value algebraically c. To determine To explain: the right hand sideof the equation is a calculationof average value geometrically

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.6, Problem 81P

To determine

To Prove: limn→∞∑k=1nf(xk)n=1b−a∫abf(x)dx using the definition of definite integral as a limit of Riemann sums

To determine

To Explain: the left hand sideof the equation is a calculationof average value algebraically

To determine

To explain: the right hand sideof the equation is a calculationof average value geometrically

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Chapter 4.6, Problem 80P

Chapter 4.6, Problem 82P

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To Prove: lim n → ∞ ∑ k = 1 n f ( x k ) n = 1 b − a ∫ a b f ( x ) d x using the definition of definite integral as a limit of Riemann sums

Solution Summary: The author explains how the accumulation function of the area A(x) can be written using the definition of definite integral.

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To Prove: limn→∞∑k=1nf(xk)n=1b−a∫abf(x)dx using the definition of definite integral as a limit of Riemann sums

To Explain: the left hand sideof the equation is a calculationof average value algebraically

To explain: the right hand sideof the equation is a calculationof average value geometrically

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Chapter 4 Solutions