Explanation Riemann Sum is defined as the approximation of the integral by computing the finite sum. Given: Given the details on the approximate height at quarter-meter intervals of a mound of gravel. Formula: If the interval [ a , b ] is divided into n sub intervals, then the length of each side is given as Δ x = b − a n Riemann Sum is given as S = ∑ i = 1 n ∑ j = 1 m f ( x i ) Δ x i Calculation: Table is given as follows: We can observe that each sub rectangle is a square, of side 0.25 , i.e. we have Δ x = 0.25 Δ y = 0.25 Thus, Area of sub rectangle, Δ A = ( Δ x ) ( Δ y ) Δ A = ( 0.25 ) ( 0.25 ) Δ A = 0 .0625 Let us first compute the sum using lower left vertices: The lower left vertices of sub rectangles are the below points P 11 = ( 0 , 0 ) P 21 = ( 0.25 , 0 ) P 31 = ( 0.50 , 0 ) P 41 = ( 0.75 , 0 ) P 12 = ( 0 , 0 .25 ) P 22 = ( 0.25 , 0 .25 ) P 32 = ( 0.50 , 0 .25 ) P 42 = ( 0.75 , 0 .25 ) P 13 = ( 0 , 0 .50 ) P 23 = ( 0 .25, 0 .50 ) P 33 = ( 0 .50, 0 .50 ) P 43 = ( 0 .75, 0 .50 ) Compute the value of the function f ( x , y ) at the above points f ( P 11 ) = f ( 0 , 0 ) = 0.1 f ( P 21 ) = f ( 0.25 , 0 ) = 0.15 f ( P 12 ) = f ( 0 , 0 .25 ) = 0.15 f ( P 22 ) = f ( 0.25 , 0 .25 ) = 0.2 f ( P 13 ) = f ( 0 , 0 .50 ) = 0.2 f ( P 23 ) = f ( 0.25 , 0 .50 ) = 0.3 f ( P 31 ) = f ( 0.50 , 0 ) = 0.2 f ( P 41 ) = f ( 0.75 , 0 ) = 0.15 f ( P 32 ) = f ( 0.50 , 0 .25 ) = 0.4 f ( P 42 ) = f ( 0.75 , 0 .25 ) = 0.3 f ( P 33 ) = f ( 0.50 , 0 .50 ) = 0.5 f ( P 43 ) = f ( 0.75 , 0 .50 ) = 0.4 Riemann Sum is given as follows: S 4 , 3 = ∑ i = 1 4 ∑ j = 1 3 f ( P i , j ) Δ A S 4 , 3 = 0...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Estimation

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Chapter 16.1, Problem 11E

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Given below provides the detail on the approximate height at quarter-meter intervals of a mound of gravel. Find the volume of the mound by computing the average of the two Riemann’s Sums $S_{4, 3}$ with lower-left and upper-left vertices of the sub rectangles as sample points

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Chapter 16.1, Problem 10E

Chapter 16.1, Problem 12E

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Chapter 16 Solutions

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Given below provides the detail on the approximate height at quarter-meter intervals of a mound of gravel. Find the volume of the mound by computing the average of the two Riemann’s Sums S 4 , 3 with lower-left and upper-left vertices of the sub rectangles as sample points

Solution Summary: The author explains Riemann Sum as the approximation of the integral by computing the finite sum.

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Given below provides the detail on the approximate height at quarter-meter intervals of a mound of gravel. Find the volume of the mound by computing the average of the two Riemann’s Sums $S_{4, 3}$ with lower-left and upper-left vertices of the sub rectangles as sample points

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Chapter 16 Solutions

Given below provides the detail on the approximate height at quarter-meter intervals of a mound of gravel. Find the volume of the mound by computing the average of the two Riemann’s Sums S 4 , 3 with lower-left and upper-left vertices of the sub rectangles as sample points

Solution Summary: The author explains Riemann Sum as the approximation of the integral by computing the finite sum.

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Given below provides the detail on the approximate height at quarter-meter intervals of a mound of gravel. Find the volume of the mound by computing the average of the two Riemann’s Sums S4,3with lower-left and upper-left vertices of the sub rectangles as sample points

Want to see the full answer?

Chapter 16 Solutions

Given below provides the detail on the approximate height at quarter-meter intervals of a mound of gravel. Find the volume of the mound by computing the average of the two Riemann’s Sums $S_{4, 3}$ with lower-left and upper-left vertices of the sub rectangles as sample points