Explanation Given: f ( x ) = x 2 on the interval [ 2 , 5 ] . Formula used: Let a closed interval [ a , b ] be partitioned by points a < x 1 < x 2 < x 3 ... < x n − 1 < b , where the lengths of the resulting intervals between the points are denoted Δ x 1 , Δ x 2 , Δ x 3 , ... , Δ x n . Let c j be an arbitrary point in the j th subinterval. Then the quantity ∑ j = 1 n f ( c j ) Δ x j is called a Riemann sum for a given function f ( x ) . Calculation: Two Riemann sums for f ( x ) = x 2 on the interval [ 2 , 5 ] will vary. Here are two possibilities. Take N = 5 , P = { x 0 = 2 , x 1 = 2.7 , x 2 = 3.1 , x 3 = 3.6 , x 4 = 4.2 , x 5 = 5 } and C = { c 1 = 2.5 , c 2 = 3 , c 3 = 3.5 , c 4 = 4 , c 5 = 4.5 } Then, R ( f , P , C ) = ∑ j = 1 5 Δ x j f ( c j ) = ( 2.7 − 2 ) ( 2.5 ) 2 + ( 3.1 − 2.7 ) ( 3 ) 2 + ( 3.6 − 3.1 ) ( 3.5 ) 2 + ( 4...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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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 5, Problem 12CRE

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Any two Riemann sums for $f (x) = x^{2}$ on the interval $[2, 5]$ , choose partitions with at least five subintervals of unequal widths and intermediate points that are neither endpoints nor midpoints.

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Chapter 5, Problem 11CRE

Chapter 5, Problem 13CRE

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Any two Riemann sums for f ( x ) = x 2 on the interval [ 2 , 5 ] , choose partitions with at least five subintervals of unequal widths and intermediate points that are neither endpoints nor midpoints.

Solution Summary: The author explains how two Riemann sums for f (x ) = x 2 on the interval are 39.9 and 38.345.

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Any two Riemann sums for $f (x) = x^{2}$ on the interval $[2, 5]$ , choose partitions with at least five subintervals of unequal widths and intermediate points that are neither endpoints nor midpoints.

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Chapter 5 Solutions

Any two Riemann sums for f ( x ) = x 2 on the interval [ 2 , 5 ] , choose partitions with at least five subintervals of unequal widths and intermediate points that are neither endpoints nor midpoints.

Solution Summary: The author explains how two Riemann sums for f (x ) = x 2 on the interval are 39.9 and 38.345.

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Any two Riemann sums for f(x)=x2 on the interval [2,5], choose partitions with at least five subintervals of unequal widths and intermediate points that are neither endpoints nor midpoints.

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Chapter 5 Solutions

Any two Riemann sums for $f (x) = x^{2}$ on the interval $[2, 5]$ , choose partitions with at least five subintervals of unequal widths and intermediate points that are neither endpoints nor midpoints.