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: The definite integral is ∫ 0 4 [ ∫ x 2 3 x + y 2 d y ] d x . From the definition, the order of integration is regular y region. That is, R = { ( x , y ) | x ≤ y ≤ 2 , 0 ≤ x ≤ 4 } . Use the online graphing calculator to sketch the graph of the region x ≤ y ≤ 2 , 0 ≤ x ≤ 4 as shown in below Figure 1. From Figure 1, observe that the region of integration can be written as regular x region. That is, R = { ( x , y ) | 0 ≤ x ≤ y 2 , 0 ≤ y ≤ 2 } . Therefore, the reverse order of integration is ∫ 0 2 [ ∫ 0 y 2 3 x + y 2 d x ] d y . Compute the value of the iterated integral ∫ 0 2 [ ∫ 0 y 2 3 x + y 2 d x ] d y as follows. First, compute the integral with respect to x . ∫ 0 2 [ ∫ 0 y 2 3 x + y 2 d x ] d y = ∫ 0 2 [ 1 3 ∫ 0 y 2 u d u ] d y [ Let u = 3 x + y 2 then d u = 3 d x ] = ∫ 0 2 [ 1 3 [ u 3 2 ( 3 2 ) ] ] 0 y 2 d y = ∫ 0 2 [ 1 3 [ 2 u 3 2 3 ] ] 0 y 2 d y = ∫ 0 2 [ 2 u 3 2 9 ] 0 y 2 d y = ∫ 0 2 [ 2 ( 3 x + y 2 ) 3 2 9 ] 0 y 2 d y Apply the limit value for x

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

13th Edition

ISBN: 9780321869838

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

Publisher: PEARSON

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Chapter 7.7, Problem 40E

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To find: The reverse order of the integration ∫04[∫x23x+y2dy]dx then to evaluate it.

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Chapter 7.7, Problem 39E

Chapter 7.7, Problem 41E

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Chapter 7 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

Ch. 7.1 - Prob. 1MP Ch. 7.1 - Matched Problem 2 Find f(2, 2, 3) for f in Example...Ch. 7.1 - Prob. 3MP Ch. 7.1 - Matched Problem 4 For the box in Example 4, find...Ch. 7.1 - Matched Problem 5 Refer to Example 5. Find the...Ch. 7.1 - Matched Problem 6 Find the coordinates of the...Ch. 7.1 - Matched Problem 7 (A) Describe the cross sections...Ch. 7.1 - Explore and Discuss 1 Imagine that you are facing...Ch. 7.1 - In Problems 18, find the indicated value of the...Ch. 7.1 - Prob. 2E

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The reverse order of the integration ∫ 0 4 [ ∫ x 2 3 x + y 2 d y ] d x then to evaluate it.

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To find: The reverse order of the integration ∫04[∫x23x+y2dy]dx then to evaluate it.

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Chapter 7 Solutions