Explanation Given data: y The integral ∫ − 2 3 ( 4 − x 2 ) d x and the interval of integration [ − 2 , 3 ] . 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 k = n ( n + 1 ) 2 and ∑ k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) 6 Proof: We have f ( x ) = 4 − x 2 and the interval of integration [ a , b ] = [ − 2 , 3 ] . Therefore, △ x = 3 − ( − 2 ) n = 5 n and taking right most point of interval [ x k − 1 , x k ] , i.e., x k = a + k △ x = − 2 + k 5 n = − 2 + 5 k n . ⇒ f ( x k ) = 4 − x 2 = 4 − ( − 2 + 5 k n ) 2 = 4 − ( 4 + 25 n 2 k 2 − 20 n k ) = 4 − 4 − 25 n 2 k 2 + 20 n k ⇒ f ( x k ) = − 25 n 2 k 2 + 20 n k 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 ∫ − 2 3

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

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Chapter 4, Problem 15SC

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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 14SC

Chapter 4, Problem 16SC

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Calculus

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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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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 Solutions