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 ) = 1 ; x 2 + y 2 + z 2 ≤ 4 z , z ≥ x 2 + y 2 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 , z ≥ x 2 + y 2 Consider the equation: x 2 + y 2 + z 2 = 4 z Now, x 2 + y 2 + z 2 − 4 z = 0 x 2 + y 2 + z 2 − 4 z + 4 = 4 x 2 + y 2 + ( z − 2 ) 2 = 4 This is the sphere with radius 2 and center at ( 0 , 0 , 2 ) . Now, to find the limits of the integral, using the spherical co-ordinates in the equation z = x 2 + y 2 and x 2 + y 2 + z 2 = 4 z , we get x 2 + y 2 = z 2 ( ρ sin φ cos θ ) 2 + ( ρ sin φ sin θ ) 2 = ρ 2 cos 2 φ ρ 2 sin 2 φ ( cos 2 θ + sin 2 θ ) = ρ 2 cos 2 φ ρ 2 sin 2 φ = ρ 2 cos 2 φ ( S i n c e sin 2 θ + cos 2 θ = 1 )

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.4, Problem 48E

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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 47E

Chapter 16.4, Problem 49E

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Chapter 16 Solutions

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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.