Explanation Given info. f ( x , y , z ) = x y z W : 0 ≤ z ≤ y ≤ x ≤ 1 Concept used: Identify the bounded region, find out the limit of variable using this diagram. Calculation: Given inequality 0 ≤ z ≤ y ≤ x ≤ 1 Fromthe above inequality z ≤ y ≤ x Take the projection of bounded region on z x plane Inclined line is intersection of z = y → ( 1 ) y = x → ( 2 ) From equation ( 1 ) and (2) z = x From the diagram z varies 0 ≤ z ≤ x And x varies 0 ≤ x ≤ 1 0 ≤ x ≤ 1 , 0 ≤ z ≤ x , z ≤ y ≤ x Find the value of ∭ W d v d v = d y d z d x ( and it is order of integration ) = ∫ 0 1 ∫ 0 x ∫ z x d z d y d x Integrating w.r.t z assuming y and x as constant = ∫ 0 1 ∫ 0 x [ ∫ z x d z ] d y d x = ∫ 0 1 ∫ 0 x [ y ] z x d y d x Putting the value of limit of y = ∫ 0 1 ∫ 0 x ( x − z ) d y d x Integrate w.r.t z , = ∫ 0 1 [ ∫ 0 x ( x − z ) d z ] d x = ∫ 0 1 [ x z − z 2 2 ] 0 x . d x Putting the value of limit of x = ∫ 0 1 ( x 2 − x 2 2 ) d x = ∫ 0 1 x 2 2 d x = [ x 3 6 ] 0 1 = 1 6 Evaluating the value = ∭ W f ( x ...

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ISBN: 9781319434540

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.3, Problem 40E

To determine

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

$f (x, y, z) = x y z$

$W : 0 \leq z \leq y \leq x \leq 1$

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Chapter 16.3, Problem 39E

Chapter 16.3, Problem 41E

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Chapter 16 Solutions

ACHIEVE STANDALONE ACCESS F/ CALC 4E

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Compute the average value of f ( x , y , z ) over the region W. f ( x , y , z ) = x y z W : 0 ≤ z ≤ y ≤ x ≤ 1

Solution Summary: The author explains how to determine the average value of f over the region W. Identify the bounded region, find out the limit of variable using this diagram.

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

$f (x, y, z) = x y z$

$W : 0 \leq z \leq y \leq x \leq 1$

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Chapter 16 Solutions

Compute the average value of f ( x , y , z ) over the region W. f ( x , y , z ) = x y z W : 0 ≤ z ≤ y ≤ x ≤ 1

Solution Summary: The author explains how to determine the average value of f over the region W. Identify the bounded region, find out the limit of variable using this diagram.

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Compute the average value of f(x,y,z)over the region W. f(x,y,z)=xyz W:0≤z≤y≤x≤1

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Chapter 16 Solutions

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

$f (x, y, z) = x y z$

$W : 0 \leq z \leq y \leq x \leq 1$