(a) Explanation The rectangular coordinates ( x , y , z ) corresponding to the cylindrical coordinates ( r , θ , z ) is, x = r cos θ y = r sin θ z = z Substitute x = r cos θ , y = r sin θ and z = z in the given problem to convert rectangular coordinates into cylindrical coordinates. Thus, the integral becomes as given below. If f is a cylindrical region E given by h 1 ( θ ) ≤ r ≤ h 2 ( θ ) , α ≤ θ ≤ β , u 1 ( r cos θ , r sin θ ) ≤ z ≤ u 1 ( r cos (b) To determine To explain: How to change rectangular coordinates to spherical coordinates in a triple integral. (c) To determine To explain: The reason to change from cylindrical coordinate system to spherical coordinate system or vice versa.

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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 15, Problem 9CC

(a)

To determine

To explain: How to change rectangular coordinates to cylindrical coordinates in a triple integral.

(b)

To determine

To explain: How to change rectangular coordinates to spherical coordinates in a triple integral.

(c)

To determine

To explain: The reason to change from cylindrical coordinate system to spherical coordinate system or vice versa.

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Chapter 15, Problem 8CC

Chapter 15, Problem 10CC

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Chapter 15 Solutions

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How to change rectangular coordinates to cylindrical coordinates in a triple integral.

Solution Summary: The author explains how to convert rectangular coordinates into cylindrical ones in a triple integral.

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To explain: How to change rectangular coordinates to cylindrical coordinates in a triple integral.

To explain: How to change rectangular coordinates to spherical coordinates in a triple integral.

To explain: The reason to change from cylindrical coordinate system to spherical coordinate system or vice versa.

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