(a) Explanation Given information: Consider the definite integral ∫ 0 1 2 x 2 d x . Formula used: Let f is continuous function defined on an interval [ a , b ] , n is a positive integer. Δ x = b − a n , x k = a + k Δ x . The right sum of n- rectangles is = ∑ k = 1 n f ( x k ) Δ x Calculation: From the given data, we have [ a , b ] = [ 0 , 1 ] Δ x = b − a n = 1 − 0 n = 1 n x k = a + k Δ x = 0 + k ( 1 / n ) = k / n The right sum is given by = ∑ k = (b) To determine To find: The value of definite integral with the given values of n. (c) To determine To find: The exact value of the given definite integral as n tends to infinite.

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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Question

Chapter 4.3, Problem 43S

(a)

To determine

To find: The formula for n- rectangle right sum of the given definite integral f on [a,b].

(b)

To determine

To find: The value of definite integral with the given values of n.

(c)

To determine

To find: The exact value of the given definite integral as n tends to infinite.

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Chapter 4.3, Problem 42S

Chapter 4.3, Problem 44S

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Calculus

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The formula for n- rectangle right sum of the given definite integral f on [ a , b ] .

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To find: The formula for n- rectangle right sum of the given definite integral f on [a,b].

To find: The value of definite integral with the given values of n.

To find: The exact value of the given definite integral as n tends to infinite.

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