Explanation Given info. f ( x , y , z ) = x y sin ( π z ) , W = [ 0 , 1 ] × [ 0 , 1 ] × [ 0 , 1 ] Concept used: Average value of function f ( x , y , z ) , over a bounded region W is defined as. 〈 f ( x , y , z ) 〉 = ∭ W f ( x , y , z ) d v ∭ W d v Here limit of variables are independent to each other so limit can perform in any order. Calculation: First calculate ∭ W d v d v = d x × d y × d z = d y × d x × d z = d z × d x × d y andso on W = [ 0 , 1 ] × [ 0 , 1 ] × [ 0 , 1 ] Perform integration in order of d x d y d z V = ∭ W d v = ∫ 0 1 ∫ 0 1 ∫ 0 1 d x d y d z (Putting the limit of variable in same as performing the integration) = ∫ 0 1 ∫ 0 1 [ ∫ 0 1 d x ] d y d z Integrate w.r.t x , assuming y and z as constant ∫ 0 1 ∫ 0 1 [ x ] 0 1 d y d z Putting the value of limit x ∫ 0 1 ∫ 0 1 [ 1 − 0 ] d y d z = ∫ 0 1 ∫ 0 1 d y d z Integrate w.r.t y . Assuming z as constant = ∫ 0 1 [ y ] 0 1 d z Putting the value of limit y = ∫ 0 1 [ 1 − 0 ] d z = ∫ 0 1 d z = [ z ] 0 1 = 1 − 0 = 1 ∴ V = ∭ W d v = 1 cubic unit Finding the value of ∭ W f ( x , y , z ) d v ( ∵ d v = d x d y d z ∴ Order of integration = d x d y d z ⇒ Putting limit in the same order ) ∫ 0 1 ∫ 0 1 ∫ 0 1 x y sin ( π z ) d x d y d z Performing the integration w

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.3, Problem 39E

To determine

The average value of f(x,y,z) over the region W.

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Chapter 16.3, Problem 38E

Chapter 16.3, Problem 40E

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Chapter 16 Solutions

CALCULUS (CLOTH)

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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The average value of f ( x , y , z ) over the region W.

Solution Summary: The author calculates the average value of f(x,y,z) over a bounded region W.

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The average value of f(x,y,z) over the region W.

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Chapter 16 Solutions