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

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.4, Problem 49E

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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 48E

Chapter 16.4, Problem 50E

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Chapter 16 Solutions

CALCULUS (CLOTH)

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 (x, y, and z) 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 (x, y, and z) 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.