If is a three-dimensional region, then its volume or its total mass can be computed by one of the integrals in Table 8.3.1. If the density in is 1, the integrals yield the volume of ; otherwise, they yield the total mass in .
Cartesian
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Cylindrical
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Spherical
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Table 8.3.1 Total volume or mass in three-dimensional region
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If the triple integrals in Tables 8.3.1 and 8.3.2 are iterated in Cartesian coordinates, is one of the six orderings of the differentials ; in cylindrical coordinates, is one of the six orderings of the differentials ; and in spherical coordinates, is one of the six orderings of the differentials .
Table 8.3.2 lists the integrals whose values are the first moments for a three-dimensional region
First Moment
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Cartesian
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Cylindrical
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Spherical
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Table 8.3.2 First moments for calculating a centroid () or a center of mass
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For a three-dimensional region , Table 8.3.3 provides the Cartesian coordinates of either the centroid or center of mass.
Coordinate
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Centroid
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Center of Mass
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Table 8.3.3 Centroid or Center of Mass
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