Explanation Given into. f ( x , y , z ) = y 2 The region within the cylinder x 2 + y 2 = 4 , where 0 ≤ z ≤ y . Concept used: To decide the limit of x , y , z draw the graph of x 2 + y 2 = 4 , z = y Take the projection of above region on x − y plane. From above figure decide the limit of x and y 0 ≤ y ≤ 4 − x 2 − 2 ≤ x ≤ 2 Calculation: ∭ w f ( x , y , z ) d v W: 0 ≤ y ≤ 4 − x 2 , 0 ≤ z ≤ y f ( x , y , z ) = y 2 Putting the value of function and limit in same order as you will perform integration but should be meaningful. ∫ − 2 2 ∫ 0 4 − x 2 ∫ 0 y y 2 d z . d y . d x First perform integration w.r.t z. = = ∫ − 2 2 ∫ 0 4 − x 2 [ ∫ 0 y y 2 d z ] d y . d x = ∫ − 2 2 ∫ 0 4 − x 2 [ y 2 d z ] 0 y d y . d x putting the value of limit z = ∫ − 2 2 ∫ 0 4 − x 2 y 3 d y . d x performing integratio w ...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.3, Problem 18E

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Compute the integral of $f (x, y, z) = y^{2}$ over the region within the cylinder $x^{2} + y^{2} = 4$ , where $0 \leq z \leq y$ .

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Chapter 16.3, Problem 17E

Chapter 16.3, Problem 19E

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Chapter 16 Solutions

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Compute the integral of f ( x , y , z ) = y 2 over the region within the cylinder x 2 + y 2 = 4 , where 0 ≤ z ≤ y .

Solution Summary: The author calculates the integral of f (x, y, and z) = Y 2 over the region within the cylinder.

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Compute the integral of $f (x, y, z) = y^{2}$ over the region within the cylinder $x^{2} + y^{2} = 4$ , where $0 \leq z \leq y$ .

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Chapter 16 Solutions

Compute the integral of f ( x , y , z ) = y 2 over the region within the cylinder x 2 + y 2 = 4 , where 0 ≤ z ≤ y .

Solution Summary: The author calculates the integral of f (x, y, and z) = Y 2 over the region within the cylinder.

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Compute the integral of f(x,y,z)=y2over the region within the cylinder x2+y2=4, where 0≤z≤y.

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Chapter 16 Solutions

Compute the integral of $f (x, y, z) = y^{2}$ over the region within the cylinder $x^{2} + y^{2} = 4$ , where $0 \leq z \leq y$ .