Explanation Definition used: The average value of the function is 1 ( b − a ) ( d − c ) ∬ f ( x , y ) d A over the rectangle. Calculation: Given that the region is R = { ( x , y ) | − 1 ≤ x ≤ 2 , 1 ≤ y ≤ 4 } . Consider the integral as follows. 1 ( b − a ) ( d − c ) ∬ R f ( x , y ) d A = 1 ( 2 − ( − 1 ) ) ( 4 − 1 ) ∬ R x 2 + y 2 d A = 1 9 ∬ R x 2 + y 2 d A Compute the integral with respect to y . 1 9 ∫ − 1 2 ∫ 1 4 x 2 + y 2 d y d x = 1 9 ∫ − 1 2 [ ∫ 1 4 ( x 2 + y 2 ) d y ] d x = 1 9 ∫ − 1 2 [ ∫ 1 4 ( x 2 d y + y 2 d y ) ] d x = 1 9 ∫ − 1 2 [ x 2 ∫ 1 4 d y + ∫ 1 4 y 2 d y ] d x = 1 9 ∫ − 1 2 [ x 2 [ y ] 1 4 + [ y 3 3 ] 1 4 ] d x Apply the limit value for y

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 7.6, Problem 28E

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Chapter 7.6, Problem 27E

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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 average value of f ( x , y ) = x 2 + y 2 .

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