Explanation Definitions used: Regular x region: “A region R in the xy plane is a regular x region if there exist functions f ( x ) and g ( x ) and numbers a and b such that R = { ( x , y ) | g ( x ) ≤ y ≤ f ( x ) , a ≤ x ≤ b } ”. Regular y region: “A region R in the xy plane is a regular y region if there exist functions h ( y ) and k ( y ) and numbers c and d such that R = { ( x , y ) | h ( y ) ≤ x ≤ k ( y ) , c ≤ y ≤ d } ”. Calculation: Compute the integral ∫ 0 2 [ ∫ 0 y ( y − x ) 4 d x ] d y with respect to x . Let u = y − x , then the derivative of u is d u = − d x . Compute the integral with respect to u . ∫ 0 2 [ ∫ 0 y ( y − x ) 4 d x ] d y = ∫ 0 2 [ − 1 ∫ 0 y u 4 d u ] d y = − 1 ∫ 0 2 [ u 5 5 ] 0 y d y = − 1 ∫ 0 2 [ ( y − x ) 5 5 ] 0 y d y = − 1 5 ∫ 0 2 [ ( y − x ) 5 ] 0 y d y Apply the limits of x . ∫ 0 2 [ ∫ 0 y ( y − x ) 4 d x ] d y = − 1 5 ∫ 0 2 [ ( y − y ) 5 − ( y − 0 ) 5 ] 0 y d y = − 1 5 ∫ 0 2 − y 5 d y = 1 5 ∫ 0 2 y 5 d y Compute the integral with respect to y . ∫ 0 2 [ ∫ 0 y ( y − x ) 4 d x ] d y = 1 5 [ y 6 6 ] 0 2 = [ y 6 30 ] 0 2 Apply the limits of y . ∫ 0 2 [ ∫ 0 y ( y − x ) 4 d x ] d y = [ 2 6 30 − 0 ] = 64 30 = 32 15 Thu, the value of the given iterated integral is 32 15 . Use the online graphing calculator to sketch the graph of 0 ≤ x ≤ y , 0 ≤ y ≤ 2 as shown in below Figure 1. From Figure 1, it is observed that the shaded region is R = { ( x , y ) | 0 ≤ x ≤ y , 0 ≤ y ≤ 2 }

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134674148

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…

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Chapter 14.7, Problem 34E

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To evaluate: The integral ∫02[∫0y(y−x)4dx]dy and graph the region of integration, then reverse the order of the integration and do the same.

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Chapter 14.7, Problem 33E

Chapter 14.7, Problem 35E

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Chapter 14 Solutions

College Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 14.1 - Matched Problem 1 Find C(20, 10) for the cost...Ch. 14.1 - Prob. 2MP Ch. 14.1 - Prob. 3MP Ch. 14.1 - Prob. 4MP Ch. 14.1 - Prob. 5MP Ch. 14.1 - Prob. 6MP Ch. 14.1 - Prob. 7MP Ch. 14.1 - Prob. 1ED Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E

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The integral ∫ 0 2 [ ∫ 0 y ( y − x ) 4 d x ] d y and graph the region of integration , then reverse the order of the integration and do the same.

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To evaluate: The integral ∫02[∫0y(y−x)4dx]dy and graph the region of integration, then reverse the order of the integration and do the same.

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Chapter 14 Solutions