Explanation Definition used: The average value of the function that is defined over a rectangle is 1 ( b − a ) ( d − c ) ∫ a b ∫ c d f ( x , y ) d A . Given: The concentration C of particulate matter at a point d miles from the food plant is given by C = 50 − 9 d 2 . From the figure, the function over the region with − 2 ≤ x ≤ 2 and − 1 ≤ y ≤ 1 . That is, the length and breadth for the town from a square 4 and 2 miles respectively. The distance from origin to the point ( x , y ) is d = x 2 + y 2 or d 2 = x 2 + y 2 . Calculation: Consider the integral over the limits − 2 ≤ x ≤ 2 and − 1 ≤ y ≤ 1 as follows. 1 ( b − a ) ( d − c ) ∬ R C ( x , y ) d A = 1 ( 2 − ( − 2 ) ) ( 1 − ( − 1 ) ) ∬ R [ 50 − 9 ( x 2 + y 2 ) ] d A = 1 ( 4 ) ( 2 ) ∬ R [ 50 − 9 ( x 2 + y 2 ) ] d A = 1 8 ∫ − 2 2 ∫ − 1 1 [ 50 − 9 ( x 2 + y 2 ) ] d y d x Compute the integral with respect to y . 1 8 ∫ − 2 2 ∫ − 1 1 50 − 9 ( x 2 + y 2 ) d y d x = 1 8 ∫ − 2 2 [ 50 ∫ − 1 1 d y − 9 x 2 ∫ − 1 1 d y − 9 ∫ − 1 1 y 2 d y ] d x = 1 8 ∫ − 2 2 [ 50 [ y ] − 1 1 − 9 x 2 [ y ] − 1 1 − 9 [ y 3 3 ] − 1 1 ] d x = 1 8 ∫ − 2 2 [ 50 [ y ] − 1 1 − 9 x 2 [ y ] − 1 1 − 3 [ y 3 ] − 1 1 ] d x Apply the limit value for y

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Chapter 7.6, Problem 53E

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To find: The average concentration of fine particulate matter throughout town for C=50−9d2.

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Chapter 7.6, Problem 52E

Chapter 7.6, Problem 54E

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The average concentration of fine particulate matter throughout town for C = 50 − 9 d 2 .

Solution Summary: The author calculates the average concentration of fine particulate matter throughout town for C=50-9d2.

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