Explanation Given info: f ( x , y , z ) = z w: Concept used: From the figure it becomes clear that z − varies from x y − plane to upto hemisphere 0 ≤ z ≤ 3 − x 2 − y 2 Equation of sphere x 2 + y 2 + z 2 = ( 3 ) 2 , so 0 ≤ z ≤ 3 − x 2 − y 2 Projection of this on x y plane is given as D is in the x y plane and bounded by x = 1 , y = 0 and x = y So y varies from 0 to x And x varies from 0 to 1 First integrate w.r.t z then with respect to y and at last integrate w.r.t x . Calculation: ∫ ∫ W ∫ f ( x , y , z ) d v w: ( putting value of function and their limti in same order as order of integration ) = ∫ 0 1 ∫ 0 x ∫ 0 9 − x 2 − y 2 z d z d y d x Frist perform integration w .r ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 16.3, Problem 15E

To determine

Calculate the integral of $f (x, y, z) = z$ over the region $w$ in figure below the hemisphere of radius $3$ and lying over the triangle $D$ in the $x y$ plane bounded by $x = 1, y = 0 and x = y$

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Chapter 16.3, Problem 14E

Chapter 16.3, Problem 16E

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Chapter 16 Solutions

Calculus - Standalone book

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Calculate the integral of f ( x , y , z ) = z over the region w in figure below the hemisphere of radius 3 and lying over the triangle D in the x y plane bounded by x = 1 , y = 0 and x = y

Solution Summary: The author calculates the integral of f ( x, y ) = z over the region w in figure below the hemisphere of radius 3.

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Calculate the integral of $f (x, y, z) = z$ over the region $w$ in figure below the hemisphere of radius $3$ and lying over the triangle $D$ in the $x y$ plane bounded by $x = 1, y = 0 and x = y$

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Calculate the integral of f ( x , y , z ) = z over the region w in figure below the hemisphere of radius 3 and lying over the triangle D in the x y plane bounded by x = 1 , y = 0 and x = y

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Calculate the integral of $f (x, y, z) = z$ over the region $w$ in figure below the hemisphere of radius $3$ and lying over the triangle $D$ in the $x y$ plane bounded by $x = 1, y = 0 and x = y$