Explanation Given info: The solid region is bounded by two paraboloids z = 8 − x 2 − y 2 and z = x 2 + y 2 . The mass density is, f ( x , y , z ) = ( x 2 + y 2 ) 1 2 Calculation: Find the bounded region: The given region is bounded above by paraboloid z = 8 − x 2 − y 2 and below by paraboloid z = x 2 + y 2 . Their surfaces intersect. So, boundary of the projection region lies in x-y plane is x 2 + y 2 = 4. It is a circular region. Find the cylindrical coordinates. The bounded in cylindrical coordinates are, x = r cos θ , y = r sin θ and z = z and d V = d z d y d x = r d z d θ d r Where, r 2 = x 2 + y 2 The formula used to find the mass of the given portion on cylindrical coordinates is, m = ∭ f ( x , y , z ) d x d y d z Here, m is the mass and f ( x , y , z ) is the mass density in three dimensions. In cylindrical coordinate mass density is f ( r ) = r ( 1 ) Find the expression for mass in cylindrical coordinates: Find the range of r in term of cylindrical coordinates. Given, x 2 + y 2 ≤ 4 r 2 ≤ 4 r ≤ 2 So, range of r is from 0 to 2 Find the range of coordinate z : Given, x 2 + y 2 ≤ z ≤ 8 − x 2 − y 2 . Converting in cylindrical coordinate r ≤ z ≤ 8 − r So, range of z is from r to 8 − r Find the range of θ : The range of θ is from 0 to 2 π ...

4th Edition

ISBN: 9781319379421

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 15, Problem 41CRE

To determine

The mass of the solid bounded by $z = 8 - x^{2} - y^{2}$ and $z = x^{2} + y^{2}$ .

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Chapter 15, Problem 40CRE

Chapter 15, Problem 42CRE

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Chapter 15 Solutions

CALCULUS 4E (HC) W/ ACHIEVE ACCESS

Ch. 15.1 - Prob. 1PQ Ch. 15.1 - Prob. 2PQ Ch. 15.1 - Prob. 3PQ Ch. 15.1 - Prob. 4PQ Ch. 15.1 - Prob. 5PQ Ch. 15.1 - Prob. 6PQ Ch. 15.1 - Prob. 1E Ch. 15.1 - Prob. 2E Ch. 15.1 - Prob. 3E Ch. 15.1 - Prob. 4E

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Prob. 29E Ch. 15.6 - Prob. 30E Ch. 15.6 - Prob. 31E Ch. 15.6 - Prob. 32E Ch. 15.6 - Prob. 33E Ch. 15.6 - Prob. 34E Ch. 15.6 - Prob. 35E Ch. 15.6 - Prob. 36E Ch. 15.6 - Prob. 37E Ch. 15.6 - Prob. 38E Ch. 15.6 - Prob. 39E Ch. 15.6 - Prob. 40E Ch. 15.6 - Prob. 41E Ch. 15.6 - Prob. 42E Ch. 15.6 - Prob. 43E Ch. 15.6 - Prob. 44E Ch. 15.6 - Prob. 45E Ch. 15.6 - Prob. 46E Ch. 15.6 - Prob. 47E Ch. 15.6 - Prob. 48E Ch. 15.6 - Prob. 49E Ch. 15.6 - Prob. 50E Ch. 15.6 - Prob. 51E Ch. 15.6 - Prob. 52E Ch. 15 - Prob. 1CRE Ch. 15 - Prob. 2CRE Ch. 15 - Prob. 3CRE Ch. 15 - Prob. 4CRE Ch. 15 - Prob. 5CRE Ch. 15 - Prob. 6CRE Ch. 15 - Prob. 7CRE Ch. 15 - Prob. 8CRE Ch. 15 - Prob. 9CRE Ch. 15 - Prob. 10CRE Ch. 15 - Prob. 11CRE Ch. 15 - Prob. 12CRE Ch. 15 - Prob. 13CRE Ch. 15 - Prob. 14CRE Ch. 15 - Prob. 15CRE Ch. 15 - Prob. 16CRE Ch. 15 - Prob. 17CRE Ch. 15 - Prob. 18CRE Ch. 15 - Prob. 19CRE Ch. 15 - Prob. 20CRE Ch. 15 - Prob. 21CRE Ch. 15 - Prob. 22CRE Ch. 15 - Prob. 23CRE Ch. 15 - Prob. 24CRE Ch. 15 - Prob. 25CRE Ch. 15 - Prob. 26CRE Ch. 15 - Prob. 27CRE Ch. 15 - Prob. 28CRE Ch. 15 - Prob. 29CRE Ch. 15 - Prob. 30CRE Ch. 15 - Prob. 31CRE Ch. 15 - Prob. 32CRE Ch. 15 - Prob. 33CRE Ch. 15 - Prob. 34CRE Ch. 15 - Prob. 35CRE Ch. 15 - Prob. 36CRE Ch. 15 - Prob. 37CRE Ch. 15 - Prob. 38CRE Ch. 15 - Prob. 39CRE Ch. 15 - Prob. 40CRE Ch. 15 - Prob. 41CRE Ch. 15 - Prob. 42CRE Ch. 15 - Prob. 43CRE Ch. 15 - Prob. 44CRE Ch. 15 - Prob. 45CRE Ch. 15 - Prob. 46CRE Ch. 15 - Prob. 47CRE Ch. 15 - Prob. 48CRE Ch. 15 - Prob. 49CRE Ch. 15 - Prob. 50CRE Ch. 15 - Prob. 51CRE Ch. 15 - Prob. 52CRE Ch. 15 - Prob. 53CRE Ch. 15 - Prob. 54CRE Ch. 15 - Prob. 55CRE Ch. 15 - Prob. 56CRE Ch. 15 - Prob. 57CRE Ch. 15 - Prob. 58CRE Ch. 15 - Prob. 59CRE Ch. 15 - Prob. 60CRE Ch. 15 - Prob. 61CRE

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The mass of the solid bounded by z = 8 − x 2 − y 2 and z = x 2 + y 2 .

The mass of the solid bounded by z = 8−x2−y2and z = x2+y2.

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Chapter 15 Solutions

The mass of the solid bounded by $z = 8 - x^{2} - y^{2}$ and $z = x^{2} + y^{2}$ .