Explanation Given information: The triple integral in cartesian coordinate is: ∫ 0 1 ∫ 0 1 − x 2 ∫ 0 x 2 + y 2 z d z d y d x Conclusion: Convert the triple integral in cylindrical coordinate: The conversion formula for all three dimensions in cylindrical coordinates are x = r cos θ , y = r sin θ and z = z , The conversion formula for volume element is, d V = d z d y d x = r d z d θ d r Where, r 2 = x 2 + y 2 Find the range for cylindrical coordinates. Given 0 ≤ z ≤ x 2 + y 2 0 ≤ z ≤ r The range for y tells us that we have a portion of the right half of a disk of radius 1 centered at the origin. Since we are restricting y ’s to positive values it looks like we will have the quarter disk in the first quadrant. Given, 0 ≤ y ≤ 1 − x 2 The range of θ is from 0 to π 2 . Since the range of θ is from 0 to 1. Range of r is zero to 1. The triple integral in cylindrical coordinate is written as, ∫ 0 1 ∫ 0 π 2 ∫ 0 r z r d z d θ d r (1) First integrate with respect to variable z , ∫ 0 1

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16, Problem 35CRE

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The given integral in cylindrical coordinates and find the value of integration.

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Chapter 16, Problem 34CRE

Chapter 16, Problem 36CRE

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Chapter 16 Solutions

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The given integral in cylindrical coordinates and find the value of integration .

Solution Summary: The author explains the triple integral in cylindrical coordinates and the conversion formula for volume element.

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The given integral in cylindrical coordinates and find the value of integration.

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Chapter 16 Solutions