Explanation Since x 2 + y 2 + z 2 = ρ 2 , Where ρ is the radius vector . The polar coordinates are: x = ρ sin φ cos θ , y = ρ sin φ sin θ , z = ρ cos φ , d V = ρ 2 sin φ d ρ d θ d φ The integration of f ( x , y , z ) over the region W can be calculated by: ∭ W f ( x , y , z ) d V = ∫ δ γ ∫ α β ∫ a b ρ 2 sin φ f ( ρ sin φ cos θ , ρ sin φ s i n θ , ρ cos φ ) d ρ d θ d φ Where a ≤ ρ ≤ b α ≤ θ ≤ β δ ≤ φ ≤ γ Given: The following function with the region is given: f ( x , y , z ) = ρ ; x 2 + y 2 + z 2 ≤ 4 , z ≤ 1 , x ≥ 0 Formula Used: ∭ W f ( x , y , z ) d V = ∫ δ γ ∫ α β ∫ a b ρ 2 sin φ f ( ρ sin φ cos θ , ρ sin φ s i n θ , ρ cos φ ) d ρ d θ d φ Calculation: The given region is: x 2 + y 2 + z 2 ≤ 4 , z ≤ 1 , x ≥ 0 Since x 2 + y 2 + z 2 = ρ 2 , Now, to find the limits of the integral, using the spherical co-ordinates: Consider the equation: x 2 + y 2 + z 2 = 4 Now, ρ = ± 2 Here, ρ varies from 0 to 2. Consider the equation: z = 1 Now, ρ cos φ = 1 2 cos φ = 1 cos φ = 1 2 φ = ± π 3 Here, φ varies from π 3 to π ...

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.4, Problem 50E

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The triple integral of $f (x, y, z)$ over the given region using spherical coordinates.

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Chapter 16.4, Problem 49E

Chapter 16.4, Problem 51E

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Chapter 16 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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The triple integral of f ( x , y , z ) over the given region using spherical coordinates.

Solution Summary: The author calculates the triple integral of f over the given region using spherical coordinates.

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The triple integral of $f (x, y, z)$ over the given region using spherical coordinates.

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Chapter 16 Solutions

The triple integral of f ( x , y , z ) over the given region using spherical coordinates.

Solution Summary: The author calculates the triple integral of f over the given region using spherical coordinates.

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To calculate: The triple integral of f(x,y,z) over the given region using spherical coordinates.

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Chapter 16 Solutions

To calculate:

The triple integral of $f (x, y, z)$ over the given region using spherical coordinates.