Explanation Definition used: “The average value of the function over the rectangle with a ≤ x ≤ b and c ≤ y ≤ d is 1 ( b − a ) ( d − c ) ∬ f ( x , y ) d A .” Given: The concentration C of insects per square foot at a point d units from the food supply is given approximately by C = 10 − 1 10 d 2 . From the figure, the function over the region with − 8 ≤ x ≤ 8 and − 6 ≤ y ≤ 6 . That is, the length and breadth for the rectangular shape of an artificial habitat are 16 and 12 feet respectively. The distance from origin to the point ( x , y ) is d = x 2 + y 2 or d 2 = x 2 + y 2 . Calculation: From the definition, the values are a = − 8 , b = 8 , c = − 6 and d = 6 . Substitute d 2 = x 2 + y 2 in C = 10 − 1 10 d 2 and obtain C ( x , y ) = 10 − 1 10 ( x 2 + y 2 ) . The average concentration is computed below. 1 ( b − a ) ( d − c ) ∬ R C ( x , y ) d A = 1 ( 8 − ( − 8 ) ) ( 6 − ( − 6 ) ) ∬ R [ 10 − 1 10 ( x 2 + y 2 ) ] d A = 1 ( 16 ) ( 12 ) ∫ − 8 8 ∫ − 6 6 [ 10 − 1 10 ( x 2 + y 2 ) ] d y d x = 1 192 ∫ − 8 8 ∫ − 6 6 [ 10 − x 2 10 − y 2 10 ] d y d x Compute the integral with respect to y . 1 192 ∫ − 8 8 ∫ − 6 6 [ 10 − x 2 10 − y 2 10 ] d y d x = 1 192 ∫ − 8 8 [ 10 ∫ − 6 6 d y − x 2 10 ∫ − 6 6 d y − 1 10 ∫ − 6 6 y 2 d y ] d x = 1 192 ∫ − 8 8 [ 10 [ y ] − 6 6 − 1 10 x 2 [ y ] − 6 6 − 1 10 [ y 3 3 ] − 6 6 ] d x Apply the limit value for y

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

14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Chapter 14.6, Problem 51E

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To find: The average concentration of insects throughout the habitat.

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Chapter 14.6, Problem 50E

Chapter 14.6, Problem 52E

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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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