Explanation Given data: y The function f ( x ) = 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 ] . Formulae : ∑ k = 1 n 1 = n , ∑ k = 1 n k = n ( n + 1 ) 2 . Proof: We have f ( x ) = x and the interval of integration [ a , b ] . Therefore, △ x = b − a n and taking right most point of interval [ x k − 1 , x k ] , i.e., x k = a + k ( b − a ) n ⇒ f ( x k ) = ( a + k ( b − a ) n ) . Then by definition of definite integral ∫ a b f ( x ) d x = lim n → ∞ ∑ k = 1 n f ( x * k ) △ x = lim n → ∞ ∑ k = 1 n ( a + k ( b − a ) n ) ( b − a ) n ∫ a b f ( x ) d x =

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 4.3, Problem 60P

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To prove: For any real numbers a and b , ∫abx dx=12(b2−a2) .

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Chapter 4.3, Problem 59P

Chapter 4.3, Problem 61P

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To prove: For any real numbers a and b , ∫ a b x d x = 1 2 ( b 2 − a 2 ) .

Solution Summary: The author explains the definite integral of f(x) and the interval of integration.

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To prove: For any real numbers a and b , ∫abx dx=12(b2−a2) .

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