Explanation Explanation Given 1. Function f ( x , y , z ) = z ( x 2 + y 2 ) s 15. Region of integration in the three-dimensional space is given by x 2 + y 2 ≤ z ≤ 8 − ( x 2 + y 2 ) Formula 1. Equations to transform Cartesian coordinates into cylindrical polar coordinates where r ≥ 0 and 0 ≤ θ ≤ 2 π . x = r cos ( θ ) , y = r sin ( θ ) , and z 2. Volume element in cylindrical polar coordinates rdrdzd θ . Calculation The surface x 2 + y 2 = z is a cone with apex at origin. 8 − ( x 2 + y 2 ) = z is an inverted cone with apex at z = 8 and intersect the x − y plane in a disc of radius 8 . The surfaces x 2 + y 2 = z and 8 − ( x 2 + y 2 ) = z intersect at z = 4 in a circle of radius 2 . The volume of integration, V, is the region above the plane z = 4 and below the plane z = 8 between the outer surface x 2 + y 2 = z and the inner surface 8 − ( x 2 + y 2 ) = z . This region can be described using the cylindrical polar coordinates as x = r cos ( θ ) , y = r sin ( θ ) and z where 0 ≤ θ ≤ 2 π , and 4 ≤ z ≤ 8 ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on …

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Chapter 16.4, Problem 30E

To determine

To find

Volume integral of the function $f (x, y, z) = z \sqrt{(x^{2} + y^{2})}$ over the region

$x^{2} + y^{2} \leq z \leq 8 - (x^{2} + y^{2})$

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Chapter 16.4, Problem 29E

Chapter 16.4, Problem 31E

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Chapter 16 Solutions

CALCULUS (CLOTH)

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To find Volume integral of the function f ( x , y , z ) = z ( x 2 + y 2 ) over the region x 2 + y 2 ≤ z ≤ 8 − ( x 2 + y 2 )

Solution Summary: The author explains the volume integral of the function f ( x, y ) = z over the region of integration in the three-dimensional space.

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To find

Volume integral of the function $f (x, y, z) = z \sqrt{(x^{2} + y^{2})}$ over the region

$x^{2} + y^{2} \leq z \leq 8 - (x^{2} + y^{2})$

Want to see the full answer?

Chapter 16 Solutions

To find Volume integral of the function f ( x , y , z ) = z ( x 2 + y 2 ) over the region x 2 + y 2 ≤ z ≤ 8 − ( x 2 + y 2 )

Solution Summary: The author explains the volume integral of the function f ( x, y ) = z over the region of integration in the three-dimensional space.

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To find Volume integral of the function f(x,y,z)=z(x2+y2)over the region x2+y2≤z≤8−(x2+y2)

Want to see the full answer?

Chapter 16 Solutions

To find

Volume integral of the function $f (x, y, z) = z \sqrt{(x^{2} + y^{2})}$ over the region

$x^{2} + y^{2} \leq z \leq 8 - (x^{2} + y^{2})$