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 given integral with respect to x . ∫ 0 4 [ ∫ y 2 4 2 y ( 1 + 2 x y ) d x ] d y = ∫ 0 4 [ ∫ y 2 4 2 y d x + 2 y ∫ y 2 4 2 y x d x ] d y = ∫ 0 4 [ [ x ] y 2 4 2 y + 2 y [ x 2 2 ] y 2 4 2 y ] d y = ∫ 0 4 [ [ x ] y 2 4 2 y + y [ x 2 ] y 2 4 2 y ] d y Apply the limits of x . ∫ 0 4 [ ∫ y 2 4 2 y ( 1 + 2 x y ) d x ] d y = ∫ 0 4 [ [ 2 y − y 2 4 ] + y [ ( 2 y ) 2 − ( y 2 4 ) 2 ] ] d y = ∫ 0 4 [ 2 y − y 2 4 ] + y [ 4 y − ( y 4 16 ) ] d y = ∫ 0 4 [ 2 y − y 2 4 ] + [ 4 y 2 − y 5 16 ] d y Compute the integral with respect to y . ∫ 0 4 [ ∫ y 2 4 2 y ( 1 + 2 x y ) d x ] d y = 2 ∫ 0 4 y 1 2 d y − 1 4 ∫ 0 4 y 2 d y + 4 ∫ 0 4 y 2 d y − 1 16 ∫ 0 4 y 5 d y = 2 [ y 3 2 ( 3 2 ) ] 0 4 − 1 4 [ y 3 3 ] 0 4 + 4 [ y 3 3 ] 0 4 − 1 16 [ y 6 6 ] 0 4 = 4 3 [ y 3 2 ] 0 4 − 1 4 [ y 3 3 ] 0 4 + 4 [ y 3 3 ] 0 4 − 1 16 [ y 6 6 ] 0 4 Apply the limits of y . ∫ 0 4 [ ∫ y 2 4 2 y ( 1 + 2 x y ) d x ] d y = 4 3 [ 8 − 0 ] − 1 12 [ 64 − 0 ] + 4 3 [ 64 − 0 ] − 1 96 [ 4 , 096 − 0 ] = 32 3 − 16 3 + 256 3 − 128 3 = 160 3 = 48 Thus, the value of the given iterated integral is 48 . Use the online graphing calculator to sketch the graph of y 2 4 ≤ y ≤ 2 y , 0 ≤ y ≤ 4 as shown in below Figure 1. From Figure 1, it is observed that the shaded region is R . From the definition of regular x region it is clear that, the below defined order of integration indicates that the region is regular x region

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14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Numerical Integration

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Chapter 14.7, Problem 38E

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To evaluate: The integral ∫04[∫y242y(1+2xy)dx]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 37E

Chapter 14.7, Problem 39E

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Chapter 14 Solutions

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

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 4 [ ∫ y 2 4 2 y ( 1 + 2 x y ) 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 ∫04[∫y242y(1+2xy)dx]dy and graph the region of integration, then reverse the order of the integration and do the same.

Want to see the full answer?

Chapter 14 Solutions