Chapter 7: Triple Integration
Section 7.3: Regions with Curved Boundaries
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Essentials
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Table 7.3.1 shows the six possible iterations of a triple integral given in Cartesian coordinates. Lower-case letters are used for lower limits of integration; upper-case for upper limits. (The names used for the functions in the limits of integration pertain to just the cell in which a particular iteration is displayed. Thus, in one cell the function might appear, while in another might appear. The function name is pertinent only to the cell in which it appears.)
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Table 7.3.1 In Cartesian coordinates, the six iterations of a triple integral
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Table 7.3.2 lists the basic Maple tools for iterating a triple integral in Cartesian coordinates. It is more extensive than the similar Table 7.2.2, which lists tools for iterating triple integrals over a three-dimensional "box."
, the iterated triple-integral template in the Calculus palette
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The MultiInt command in the Student MultivariateCalculus package
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For iteration in the order :
The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Cartesian 3-D
This command is also available through the Context Panel once the Student MultivariateCalculus package has been loaded.
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The four relevant task templates
at
Tools≻Tasks≻Browse: Calculus - Vector≻Integration≻Multiple Integration≻3-D
Over a Cube, Over a Sphere, Over a Tetrahedron, Over a General 3-D Region
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The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 3-D
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In the Student VectorCalculus package, the modified int command with predefined regions:
Parallelepiped, Sphere, Tetrahedron, Region
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The Int and int commands at top-level
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Table 7.3.2 Maple tools for iterating a triple integral in Cartesian coordinates
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Examples
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For given in Examples 7.3.(13 - 20), implement an appropriate iteration of the triple integral .
Example 7.3.13
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is that portion of the first-octant lying under , between the planes and .
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Example 7.3.14
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is that portion of the first-octant lying under the plane .
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Example 7.3.15
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is the region bounded above by and below by .
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Example 7.3.16
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is the region bounded above by and below by , and lying inside the cylinder .
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Example 7.3.17
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is the region bounded above by and below by , and bounded by the planes , , , .
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Example 7.3.18
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is that portion of the first-octant bounded by the coordinate planes, , and .
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Example 7.3.19
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is the region bounded above by and below by .
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Example 7.3.20
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is that portion of the first-octant bounded above by , and on the right by .
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