Section 5.6 details changing coordinates in a double integral. Extending this discussion to the triple integral leads to the "formula"
=
|
|
|
where is a region in Cartesian -space, is its image under the invertible mapping defined by the equations
|
|
|
|
and
=
|
|
|
is the Jacobian of the transformation from to .
In particular, for cylindrical coordinates defined by the equations
|
|
|
the relevant Jacobian is . (See Example 7.4.15 for the explicit calculation of this Jacobian.)
Table 7.4.1, analogous to Table 7.3.1 for Cartesian coordinates, lists the six possible iterations for a triple integral in cylindrical coordinates.
|
|
|
|
|
|
Table 7.4.1 In cylindrical coordinates, the six iterations of a triple integral
|
|
|
As in Table 7.3.1, 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.
Table 7.4.2 lists the basic Maple tools for iterating a triple integral in Cartesian coordinates.
, the iterated triple-integral template in the Calculus palette
The Jacobian must be included in the integrand.
|
In the Student MultivariateCalculus package, the MultiInt command with the option "coordinates = cylindrical" if the coordinate names are or "coordinates = cylindrical[]" if the coordinate names are .
|
The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical
|
For iteration in the order :
The task template at Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Cylindrical
|
The Int and int commands at top-level
The Jacobian must be included in the integral
|
Table 7.4.2 Maple tools for iterating a triple integral in cylindrical coordinates
|
|
|
Table 7.4.3 illustrates the syntax by means of which the plot3d command will graph a surface defined in cylindrical coordinates.
Explicit
|
|
Parametric
|
|
Table 7.4.3 Syntax by which the plot3d command will graph in cylindrical coordinates
|
|
|
In the explicit case, the first range and its coordinate name is interpreted as the angle ; the second, as . The function is interpreted as .
In the parametric case, the parameters and could be any two of . For example, if and , then the parametric list would be .