Explanation Given data: y The integral ∫ 1 4 ( 3 x + 1 ) 2 d x and the interval of integration [ a , b ] . Concept used: Definite Integral of a Function on an Interval: Let f ( x ) be a function defined on an interval [ a , b ] . The definite integral of f ( x ) from x = a to x = b is defined to be the number ∫ a b f ( x ) d x = lim n → ∞ ∑ k = 1 n f ( x * k ) △ x provided this limit exists for △ x = b − a n , x k = a + k △ x , and x * k is any choice of point in [ x k − 1 , x k ] . The sum S n = lim n → ∞ ∑ k = 1 n f ( x * k ) △ x is called the Riemann sum. Formulae : ∑ k = 1 n 1 = n , ∑ k = 1 n k = n ( n + 1 ) 2 , and ∑ k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) 6 Proof: We have f ( x ) = ( 3 x + 1 ) 2 and the interval of integration [ a , b ] = [ 1 , 4 ] . Therefore, △ x = 4 − 1 n = 3 n and taking right most point of interval [ x k − 1 , x k ] , i.e., x k = 1 + k 3 n . Therefore, f ( x k ) = ( 3 x k + 1 ) 2 = 9 x k 2 + 6 x k + 1 . ⇒ f ( x k ) = 9 ( 1 + k 3 n ) 2 + 6 ( 1 + k 3 n ) + 1 = 9 + 81 k 2 n 2 + 54 k n + 6 ( 1 + k 3 n ) + 1 ⇒ f ( x k ) = 81 k 2 n 2 + 72 k n + 16 = 81 k 2 n 2 + 72 k n + 16 Then by definition of definite integral ∫ a b f ( x ) d x = lim n → ∞

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4, Problem 16SC

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To calculate:The exact value of given definite integral by setting up a general Riemann sum and taking limit n→∞ .

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Chapter 4, Problem 15SC

Chapter 4, Problem 17SC

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The exact value of given definite integral by setting up a general Riemann sum and taking limit n → ∞ .

Solution Summary: The author calculates the exact value of given definite integral by setting up a general Riemann sum and taking limit nto infty

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