Explanation Given : S N , N , N = 1 N 3 ∑ i = 1 N ∑ j = 1 N ∑ k = 1 N f ( i N , j N , k N ) , f ( x , y , z ) = sin ( x y z ) , N = 10 , 20 , a n d 30

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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 16.3, Problem 44E

To determine

To calculate: Using the CAS, the Riemann sum approximation $S_{N, N, N} = \frac{1}{N^{3}} \sum_{i = 1}^{N} \sum_{j = 1}^{N} \sum_{k = 1}^{N} f (\frac{i}{N}, \frac{j}{N}, \frac{k}{N})$ for $f (x, y, z) = \sin (x y z)$ for $N = 10, 20, a n d 30$ , the value of I= $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} f (x, y, z) d V$ and the estimated value of the error $| I - S_{N, N, N} |$ .

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Chapter 16.3, Problem 43E

Chapter 16.3, Problem 45E

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Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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Solution Summary: The author calculates the Riemann sum approximation S N, N and the estimated value of the error.

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To calculate: Using the CAS, the Riemann sum approximation SN,N,N=1N3∑i=1N∑j=1N∑k=1Nf(iN,jN,kN) for f(x,y,z)=sin(xyz) for N=10, 20, and 30, the value of I= ∫01∫01∫01f(x,y,z)dV and the estimated value of the error |I−SN,N,N|.

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Chapter 16 Solutions