Multiple Integration Maplet - Maple Help

Multiple Integration Maplet

 Description This template provides simplified access to the int command as modified in the VectorCalculus packages. Pressing the "Multiple Integration Maplet" button launches a maplet that computes the integral of a multivariate function over a predefined or a user-specified region.



 Multiple Integration Maplet To launch the Multiple Integration Maplet, click the button to the right. An abbreviated Help file is available from within the Maplet. Extended help is available in the attached collapsed section. Inserting Minimal Content excludes the Extended Help section. Inserting Default Content includes the Extended Help section.



Extended Help



The Multiple Integration Maplet provides simplified access to the int command as modified in the VectorCalculus packages.  These modifications allow int to create multiple (iterated) integrals, both active and inert.  A number of pre-defined regions of integration are known to the command, which also admits user-defined domains.



The Maplet contains three panels.  The initial panel is shown in Figure 1.

 Figure 1   Initial panel for the Multiple Integration Maplet

An integrand is entered in the field designated as "Function =".

By default, the Maplet expects a triple integral, but this can be changed by clicking on the 2D button to the right of "Dimensions."

The variables of integration are entered in the fields to the right of  $\left[x,y,z\right]=$ (or $\left[x,y\right]=$in the 2D case).  Thus, if integration is over a disk, the underlying int command will switch to polar coordinates, with the first variable being the radial coordinate.  (For such cases, int will automatically include the appropriate Jacobian.)

Regions of integration are accessed by clicking on the "Edit 3D Region" or "Edit 2D Region" buttons.  The panels that open are shown in Figures 2 and 3, respectively.



 Figure 2   Regions of integration for triple integrals Figure 3   Regions of integration for double integrals

Three pre-defined 3D regions are recognized, namely, the sphere, parallelepiped, and the tetrahedron.  The general 3D region is entered by providing appropriate bounding functions in the section designated by "3D Region."

Four pre-defined 2D regions are recognized, namely, the triangle, rectangle, circle, and the ellipse.  Clearly, by "circle" is meant the closure, or more appropriately, the disk.  Both the circle and ellipse can be subdivided by choosing the Sector option.  The general 2D region is entered by providing appropriate bounding functions in the section designated by "2D Region."