Explanation Given: The function is f ( x , y , z ) = z ( x 2 + y 2 + z 2 ) − 3 2 and x 2 + y 2 + z 2 ≤ 16 defined by z ≥ 2 . Formula used: In spherical coordinates, x = ρ sin ϕ c o s θ y = ρ sin ϕ sin θ z = ρ cos ϕ Calculation: Using spherical coordinates, x = ρ sin ϕ c o s θ y = ρ sin ϕ sin θ z = ρ cos ϕ So; ρ 2 = x 2 + y 2 + z 2 ρ 2 = 16 ρ = 4 Thus, 2 ≤ ρ ≤ 4 , 0 ≤ θ ≤ 2 π , 0 ≤ ϕ ≤ π . Now, function in polar coordinates can be written as; f ( ρ , θ , ϕ ) = z ( x 2 + y 2 + z 2 ) − 3 2 = ρ cos ϕ ( ρ 2 ) − 3 2 = ρ cos ϕ ( ρ ) − 3 = ρ − 2 cos ϕ So, integral in polar form is; ∫ θ = 0 2 π ∫ ϕ = 0 π ∫ ρ = 2 4 ρ >&#...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.4, Problem 53E

To determine

To calculate: The integral of $f (x, y, z) = z {(x^{2} + y^{2} + z^{2})}^{\frac{- 3}{2}}$ over the part of ball $x^{2} + y^{2} + z^{2} \leq 16$ defined by $z \geq 2$ .

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Chapter 16.4, Problem 52E

Chapter 16.4, Problem 54E

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Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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The integral of f ( x , y , z ) = z ( x 2 + y 2 + z 2 ) − 3 2 over the part of ball x 2 + y 2 + z 2 ≤ 16 defined by z ≥ 2 .

Solution Summary: The author explains how the integral of f (x, y, and z) is 0 over the part of ball.

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To calculate: The integral of $f (x, y, z) = z {(x^{2} + y^{2} + z^{2})}^{\frac{- 3}{2}}$ over the part of ball $x^{2} + y^{2} + z^{2} \leq 16$ defined by $z \geq 2$ .

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Chapter 16 Solutions

The integral of f ( x , y , z ) = z ( x 2 + y 2 + z 2 ) − 3 2 over the part of ball x 2 + y 2 + z 2 ≤ 16 defined by z ≥ 2 .

Solution Summary: The author explains how the integral of f (x, y, and z) is 0 over the part of ball.

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To calculate: The integral of f(x,y,z)=z(x2+y2+z2)−32 over the part of ball x2+y2+z2≤16 defined by z≥2.

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Chapter 16 Solutions

To calculate: The integral of $f (x, y, z) = z {(x^{2} + y^{2} + z^{2})}^{\frac{- 3}{2}}$ over the part of ball $x^{2} + y^{2} + z^{2} \leq 16$ defined by $z \geq 2$ .