Explanation Given: The function is f ( x , y , z ) = z . The region E is enclosed by the paraboloid z = x 2 + y 2 and the plane z = 4 . Formula used: If f is a cylindrical region E given by h 1 ( θ ) ≤ r ≤ h 2 ( θ ) , α ≤ θ ≤ β , u 1 ( r cos θ , r sin θ ) ≤ z ≤ u 1 ( r cos θ , r sin θ ) where 0 ≤ β − α ≤ 2 π , then, ∭ E f ( x , y , z ) d V = ∫ α β ∫ h 1 ( θ ) h 2 ( θ ) ∫ u 1 ( r cos θ , r sin θ ) u 2 ( r cos θ , r sin θ ) f ( r cos θ , r sin θ , z ) r d z d r d θ (1) If g ( x ) is the function of x and h ( y ) is the function of y and k ( z ) is the function of z , then, ∫ a b ∫ c d ∫ e f g ( x ) h ( y ) k ( z ) d z d y d x = ∫ a b g ( x ) d x ∫ c d h ( y ) d y ∫ e f k ( z ) d z (2) The cylindrical coordinates ( r , θ , z ) corresponds to the rectangular coordinates ( x , y , z ) is, r = x 2 + y 2 θ = tan − 1 ( y x ) z = z Calculation: Solve the given equations, z = x 2 + y 2 z = r 2 r 2 = 4 r = 2 It is observed that r varies from 0 to 2, θ varies from 0 to 2 π and z varies from r 2 to 4. Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the value of the given triple integral is, ∭ E f ( x , y , z ) d V = ∫ 0 2 ∫ 0 2 π ∫ r 2 4 z ( r ) d z d θ d r = ∫ 0 2 ∫ 0 2 π ∫ r 2 4 z r d z d θ d r Use the equation (2) to integrate and apply the limit values

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 12.6, Problem 18E

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To evaluate: The given triple integral by using cylindrical coordinates.

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Chapter 12.6, Problem 17E

Chapter 12.6, Problem 19E

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Essential Calculus: Early Transcendentals

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The given triple integral by using cylindrical coordinates.

Solution Summary: The author evaluates the given triple integral by using cylindrical coordinates. The value of the function is f(x,y,z)=z.

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To evaluate: The given triple integral by using cylindrical coordinates.

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Chapter 12 Solutions