Explanation Given info: W is the portion of half-cylinder. x 2 + y 2 ≤ 4 , x ≥ 0 and 0 ≤ z ≤ 3 y . The mass density ρ ( x , y , z ) of the given portion is z 2 . Formula used: The conversion formula for 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 = ∭ ρ ( x , y , z ) d x d y d z Here, m is the mass and ρ ( x , y , z ) is the mass density in three dimensions. Calculation: In order to find the expression for mass in cylindrical coordinates, let us first 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 As x ≥ 0 . Find the range of coordinate z : Given, 0 ≤ z ≤ 3 y . Converting in cylindrical coordinate 0 ≤ z ≤ 3 r sin θ So, range of z is from 0 to 3 r sin θ Find the range of θ : Given x ≥ 0 In cylindrical coordinate r cos θ ≥ 0 As cos π 2 is zero and cos 0 is 1 . The range of θ is from 0 to π 2 . Final expression for triple integral in cylindrical coordinate is, m = ∫ 0 π 2 ∫ 0 2 ∫ 0 3 r sin θ ρ ( x , y , z ) r d z d r d θ Substitute the value of mass density.

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16, Problem 42CRE

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To compute the mass of portion W using cylindrical quadrantes.

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Chapter 16, Problem 41CRE

Chapter 16, Problem 43CRE

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Chapter 16 Solutions

CALCULUS (CLOTH)

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To compute the mass of portion W using cylindrical quadrantes.

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Chapter 16 Solutions