To derive: The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable. ∭ W f ( x , y , z ) d V = ∫ θ 1 θ 2 ∫ ϕ = ϕ 1 ϕ 2 ∫ ρ = ρ 1 ρ 2 f ( ρ sin ϕ cos θ , ρ sin ϕ sin θ , ρ cos ϕ ) ρ 2 sin ϕ d ρ d ϕ d θ

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Answer 1

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 16.6, Problem 43E

To determine

To derive:

The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable.

$\underset{W}{∭} f (x, y, z) d V = \int_{θ_{1}}^{θ_{2}} \int_{ϕ = ϕ_{1}}^{ϕ_{2}} \int_{ρ = ρ_{1}}^{ρ_{2}} f (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ^{2} \sin ϕ d ρ d ϕ d θ$

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Answer 2

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 16.6, Problem 43E

To determine

To derive:

The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable.

$\underset{W}{∭} f (x, y, z) d V = \int_{θ_{1}}^{θ_{2}} \int_{ϕ = ϕ_{1}}^{ϕ_{2}} \int_{ρ = ρ_{1}}^{ρ_{2}} f (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ^{2} \sin ϕ d ρ d ϕ d θ$

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Answer 3

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 16.6, Problem 43E

To determine

To derive:

The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable.

$\underset{W}{∭} f (x, y, z) d V = \int_{θ_{1}}^{θ_{2}} \int_{ϕ = ϕ_{1}}^{ϕ_{2}} \int_{ρ = ρ_{1}}^{ρ_{2}} f (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ^{2} \sin ϕ d ρ d ϕ d θ$

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Answer 4

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 16.6, Problem 43E

To determine

To derive:

The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable.

$\underset{W}{∭} f (x, y, z) d V = \int_{θ_{1}}^{θ_{2}} \int_{ϕ = ϕ_{1}}^{ϕ_{2}} \int_{ρ = ρ_{1}}^{ρ_{2}} f (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ^{2} \sin ϕ d ρ d ϕ d θ$

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Answer 5

Chapter 16.6, Problem 43E

To determine

To derive:

The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable.

$\underset{W}{∭} f (x, y, z) d V = \int_{θ_{1}}^{θ_{2}} \int_{ϕ = ϕ_{1}}^{ϕ_{2}} \int_{ρ = ρ_{1}}^{ρ_{2}} f (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ^{2} \sin ϕ d ρ d ϕ d θ$

Answer 6

Chapter 16.6, Problem 43E

To determine

To derive:

The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable.

$\underset{W}{∭} f (x, y, z) d V = \int_{θ_{1}}^{θ_{2}} \int_{ϕ = ϕ_{1}}^{ϕ_{2}} \int_{ρ = ρ_{1}}^{ρ_{2}} f (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ^{2} \sin ϕ d ρ d ϕ d θ$

Answer 7

Chapter 16.6, Problem 43E

To determine

To derive:

The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable.

$\underset{W}{∭} f (x, y, z) d V = \int_{θ_{1}}^{θ_{2}} \int_{ϕ = ϕ_{1}}^{ϕ_{2}} \int_{ρ = ρ_{1}}^{ρ_{2}} f (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ^{2} \sin ϕ d ρ d ϕ d θ$

Answer 8

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Answer 9

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Answer 10

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Answer 11

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

Solution Summary: The author explains how the triple integral for volume in spherical coordinate system has been derived using change of variable.

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Chapter 16 Solutions

Solution Summary: The author explains how the triple integral for volume in spherical coordinate system has been derived using change of variable.

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To derive: The triple integral for calculating volume from rectangular coordinate system to spherical coordinate system using change of variable. ∭Wf(x,y,z)dV=∫θ1θ2∫ϕ=ϕ1ϕ2∫ρ=ρ1ρ2f(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)ρ2sinϕdρdϕdθ

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Chapter 16 Solutions