Explanation Given info: f ( x , y , z ) = x W: x 2 + y 2 ≤ z ≤ 4 Concept used: To decide the limit of variable you have given inequality relationship. The 3 − d region bounded by z ( x , y ) ≥ x 2 + y 2 z = 4 So W: − 2 ≤ x ≤ 2 , − 4 − x 2 ≤ y ≤ 4 − x 2 , x 2 + y 2 ≤ z ≤ 4 Calculation: W: x 2 + y 2 ≤ z ≤ 4 From the above figure it becomes clear that z = x 2 + y 2 ( Parabolic surface ) z = 4 ( is a horizontal ) Above inequality can be written as x 2 + y 2 ≤ 4 ( Because maximum value of z max =4 ) Express y as function of x y 2 ≤ 4 − x 2 Taking square root both side | y | ≤ 4 − x 2 − 4 − x 2 ≤ y ≤ 4 − x 2 Above inequality is defined only when 4 − x 2 ≥ 0 x 2 ≤ 4 − 2 ≤ x ≤ 2 Hence, W: − 2 ≤ x ≤ 2 , − 4 − x 2 ≤ y ≤ 4 − x 2 , x 2 + y 2 ≤ z ≤ 4 ∭ w f ( x ) . dv dv = d x d y d z ∫ − 2 2 ∫ − 4 − x 2 4 − x 2 ∫ x 2 + y 2 4 x d z d y d x ( Integrate with respect to z keeping x and y as constant ) = ∫ − 2 2 ∫ − 4 − x 2 4 − x 2 ( x d y d x ) ∫ x 2 + y 2 4 d z ( rewrite the expression ) (Taking constant outside the limit) = ∫ − 2 2 ∫ − 4 − x 2 4 − x 2 [ z ] x 2 + y 2 4 d y d x Putting the value of limit = ∫ − 2 2 ∫ − 4 − x 2 4 − x 2 x ( 4 − x 2 − y 2 ) d y d x (Integrate with respect to y

4th Edition

ISBN: 9781319434540

Author: Rogawski

Publisher: MAC HIGHER

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

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Question

Chapter 16.3, Problem 12E

To determine

Evaluate $\underset{w}{∭} f (x, y, z) dv$ for the function $F$ and region $W$ specified. $f (x, y, z) = x, W: x^{2} + y^{2} \leq z \leq 4$

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Chapter 16.3, Problem 11E

Chapter 16.3, Problem 13E

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Chapter 16 Solutions

ACHIEVE STANDALONE ACCESS F/ CALC 4E

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Evaluate ∭ w f ( x , y , z ) dv for the function F and region W specified. f ( x , y , z ) = x , W: x 2 + y 2 ≤ z ≤ 4

Solution Summary: The author explains how to determine the limit of variable by evaluating w f ( x, y) dv for the function F and region W.

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Evaluate $\underset{w}{∭} f (x, y, z) dv$ for the function $F$ and region $W$ specified. $f (x, y, z) = x, W: x^{2} + y^{2} \leq z \leq 4$

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Chapter 16 Solutions

Evaluate ∭ w f ( x , y , z ) dv for the function F and region W specified. f ( x , y , z ) = x , W: x 2 + y 2 ≤ z ≤ 4

Solution Summary: The author explains how to determine the limit of variable by evaluating w f ( x, y) dv for the function F and region W.

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Evaluate ∭wf(x,y,z) dv for the function Fand region Wspecified.f(x,y,z)= x , W: x2+y2≤z≤4

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Chapter 16 Solutions

Evaluate $\underset{w}{∭} f (x, y, z) dv$ for the function $F$ and region $W$ specified. $f (x, y, z) = x, W: x^{2} + y^{2} \leq z \leq 4$