To derive: The formula for volume in cylindrical coordinate system using change of variables. ∫ θ 1 θ 2 ∫ r 1 r 2 ∫ z 1 z 2 f ( r cos θ , r sin θ , z ) r d z d r 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 42E

To determine

To derive:

The formula for volume in cylindrical coordinate system using change of variables.

$\int_{θ_{1}}^{θ_{2}} \int_{r_{1}}^{r_{2}} \int_{z_{1}}^{z_{2}} f (r \cos θ, r \sin θ, z) r d z d r d θ$

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Answer 2

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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 42E

To determine

To derive:

The formula for volume in cylindrical coordinate system using change of variables.

$\int_{θ_{1}}^{θ_{2}} \int_{r_{1}}^{r_{2}} \int_{z_{1}}^{z_{2}} f (r \cos θ, r \sin θ, z) r d z d r d θ$

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Check out a sample textbook solution

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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 42E

To determine

To derive:

The formula for volume in cylindrical coordinate system using change of variables.

$\int_{θ_{1}}^{θ_{2}} \int_{r_{1}}^{r_{2}} \int_{z_{1}}^{z_{2}} f (r \cos θ, r \sin θ, z) r d z d r d θ$

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Check out a sample textbook solution

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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 42E

To determine

To derive:

The formula for volume in cylindrical coordinate system using change of variables.

$\int_{θ_{1}}^{θ_{2}} \int_{r_{1}}^{r_{2}} \int_{z_{1}}^{z_{2}} f (r \cos θ, r \sin θ, z) r d z d r d θ$

Expert Solution & Answer

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Check out a sample textbook solution

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Answer 5

Chapter 16.6, Problem 42E

To determine

To derive:

The formula for volume in cylindrical coordinate system using change of variables.

$\int_{θ_{1}}^{θ_{2}} \int_{r_{1}}^{r_{2}} \int_{z_{1}}^{z_{2}} f (r \cos θ, r \sin θ, z) r d z d r d θ$

Answer 6

Chapter 16.6, Problem 42E

To determine

To derive:

The formula for volume in cylindrical coordinate system using change of variables.

$\int_{θ_{1}}^{θ_{2}} \int_{r_{1}}^{r_{2}} \int_{z_{1}}^{z_{2}} f (r \cos θ, r \sin θ, z) r d z d r d θ$

Answer 7

Chapter 16.6, Problem 42E

To determine

To derive:

The formula for volume in cylindrical coordinate system using change of variables.

$\int_{θ_{1}}^{θ_{2}} \int_{r_{1}}^{r_{2}} \int_{z_{1}}^{z_{2}} f (r \cos θ, r \sin θ, z) r d z d r d θ$

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

To derive: The formula for volume in cylindrical coordinate system using change of variables. ∫ θ 1 θ 2 ∫ r 1 r 2 ∫ z 1 z 2 f ( r cos θ , r sin θ , z ) r d z d r d θ

Solution Summary: The author explains how to derive the formula for volume in cylindrical coordinate system using change of variables.

To derive:

The formula for volume in cylindrical coordinate system using change of variables.

$\int_{θ_{1}}^{θ_{2}} \int_{r_{1}}^{r_{2}} \int_{z_{1}}^{z_{2}} f (r \cos θ, r \sin θ, z) r d z d r d θ$

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Chapter 16 Solutions

To derive: The formula for volume in cylindrical coordinate system using change of variables. ∫ θ 1 θ 2 ∫ r 1 r 2 ∫ z 1 z 2 f ( r cos θ , r sin θ , z ) r d z d r d θ

Solution Summary: The author explains how to derive the formula for volume in cylindrical coordinate system using change of variables.