Explanation Given info. f ( x , y , z ) = e y W : 0 ≤ y ≤ 1 − x 2 , 0 ≤ z ≤ x Concept used: Identify the bounded region, find out the limit of variable using this diagram. Calculation: From the above diagram it becomes clear that bounded region W : 0 ≤ y ≤ 1 − x 2 , 0 ≤ z ≤ x 0 ≤ z ≤ x 0 ≤ y ≤ 1 − x 2 Projection of bounded region on z x plane Hence W : 0 ≤ y ≤ 1 − x 2 , 0 ≤ z ≤ x , 0 ≤ x ≤ 1 ∭ W d v = ∫ 0 1 ∫ 0 1 − x 2 ∫ 0 x d z d y d x Integrating w.r.t z assuming y and x as constant ∫ 0 1 ∫ 0 1 − x 2 [ ∫ 0 x d z ] d y d x ∫ 0 1 ∫ 0 1 − x 2 [ z ] 0 x d y d x Putting the value of limit of z ∫ 0 1 ∫ 0 1 − x 2 x d y d x (Integrating w.r.t y, assuming y as constant) ∫ 0 1 [ x y ] 0 1 − x 2 d x Putting the value of limit of y = ∫ 0 1 x ( 1 − x 2 ) d x = ∫ 0 1 ( 1 − x 3 ) d x = [ x 2 2 − x 4 4 ] 0 1 = 1 2 − 1 4 = 1 4 Evaluating the value ∭ W f ( x . y . z ) d v = ∫ 0 1 ∫ 0 1 − x 2 ∫ 0 x e 2 d z d y d x Integrating w.r.t z assume x and y as constant = ∫ 0 1 ∫ 0 1 − x 2 [ ∫ 0 x e 2 d z ] d y d x = ∫ 0 1 ∫ 0 1 − x 2 [ e y z ] 0 x d y d x Putting the limit of z = ∫ 0 1 ∫ 0 1 − x 2 e y x ...

4th Edition

ISBN: 9781319279813

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 15.3, Problem 41E

To determine

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

$f (x, y, z) = e^{y}$

$W : 0 \leq y \leq 1 - x^{2}, 0 \leq z \leq x$

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Chapter 15.3, Problem 40E

Chapter 15.3, Problem 42E

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Chapter 15 Solutions

CALCULUS:EARLY TRANS.-SAPLINGPLUS

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Compute the average value of f ( x , y , z ) over the region W. f ( x , y , z ) = e y W : 0 ≤ y ≤ 1 − x 2 , 0 ≤ z ≤ x

Compute the average value of f(x,y,z)over the region W. f(x,y,z)=ey W:0≤y≤1−x2,0≤z≤x

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Chapter 15 Solutions

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

$f (x, y, z) = e^{y}$

$W : 0 \leq y \leq 1 - x^{2}, 0 \leq z \leq x$