Explanation Given: Solid lie between x 2 + y 2 + z 2 = a 2 and x 2 + y 2 + z 2 = b 2 ; b > a and inside cone z 2 = x 2 + y 2 Formula Used: Rectangular conversion components for spherical coordinates x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ x 2 + y 2 + z 2 = ρ 2 Volume of solids in polar form ∭ V f ( x , y , z ) d V = ∭ V f ( ρ s i n ϕ c o s θ , ρ s i n ϕ s i n θ , ρ c o s ϕ ) ρ 2 sin ϕ d ρ d ϕ d θ Calculation: From given equation of sphere, x 2 + y 2 + z 2 = a 2 ρ = a …… (1) And, x 2 + y 2 + z 2 = b 2 ρ = b …… (2) Since z 2 = x 2 + y 2 will be equally inclined Therefore, ϕ limits at π 4 Bounds on ϕ are 0 ≤ ϕ ≤ π 4 Bounds on ρ are a ≤ ρ ≤ b Bounds on θ are 0 ≤ θ ≤ 2 π Volume in Spherical coordinates V = ∭ V ρ 2 sin ϕ d ρ d ϕ d θ = ∫ 0 2 π ∫ 0 π 4

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ISBN: 9780357246412

Author: Ron Larson; Bruce H. Edwards

Publisher: Cengage Limited

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Chapter 14.7, Problem 34E

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To calculate: The volume of solid lie between x2+y2+z2=a2 and x2+y2+z2=b2;b>a and inside cone z2=x2+y2 using spherical coordinates of the solid.

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Chapter 14.7, Problem 33E

Chapter 14.7, Problem 35E

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The volume of solid lie between x 2 + y 2 + z 2 = a 2 and x 2 + y 2 + z 2 = b 2 ; b > a and inside cone z 2 = x 2 + y 2 using spherical coordinates of the solid.

Solution Summary: The author calculates the volume of a solid between x+y2 +z2, and inside cone using spherical coordinates.

Videos

To calculate: The volume of solid lie between x2+y2+z2=a2 and x2+y2+z2=b2;b>a and inside cone z2=x2+y2 using spherical coordinates of the solid.

Want to see the full answer?

Chapter 14 Solutions

The volume of solid lie between x 2 + y 2 + z 2 = a 2 and x 2 + y 2 + z 2 = b 2 ; b &gt; a and inside cone z 2 = x 2 + y 2 using spherical coordinates of the solid.

Solution Summary: The author calculates the volume of a solid between x+y2 +z2, and inside cone using spherical coordinates.

Videos

To calculate: The volume of solid lie between x2+y2+z2=a2 and x2+y2+z2=b2;b>a and inside cone z2=x2+y2 using spherical coordinates of the solid.

Want to see the full answer?

Chapter 14 Solutions

The volume of solid lie between x 2 + y 2 + z 2 = a 2 and x 2 + y 2 + z 2 = b 2 ; b > a and inside cone z 2 = x 2 + y 2 using spherical coordinates of the solid.