Classroom Tips and Techniques: Visualizing Regions of Integration
Robert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow
Maplesoft

Introduction


Five of the new task templates in Maple 14 are designed to help visualize regions of integration for iterated integrals. In particular, there are task templates for double integrals in Cartesian and polar coordinates, and for triple integrals in Cartesian, cylindrical, and spherical coordinates. These task templates can be found at the end of the path
Tools_Tasks_Browse:
Calculus  Multivariate_Integration_Visualizing Regions of Integration

Each of these task templates provides for iterating the relevant multiple integral in any of its possible orders. An example for each task template can be found below.
The inspiration for these task templates lies in Maple code written in the early 1990s by Dr. Tim Murdoch while he was a faculty member at Washington and Lee University, in Lexington, Virginia. I acquired this code from Tim when he was a participant in one of the NSF workshops in computer algebra hosted by RoseHulman Institute of Technology. I used this code to great effect in my calculus classes at RHIT from then till the day I retired.
The lure of this code is that it used the same syntax in its plotting commands as Maple used in the Doubleint and Tripleint commands in the old student package. Thus, if one could draw the region of integration with one of the plotting commands, the same syntax would work in the integration commands, and vice versa.
For threedimensional regions, Tim's code rendered each of the six possible faces of the region parametrically. This is what the task templates do, but without the elegance Tim used to make the coding of each command fit the same pattern.


Visualizing Regions of Integration in 2D Cartesian Coordinates


Evaluate and Graph

Area Element

,

Value of Integral



The simplest order of integration is in which the first (inner) integration is in the direction. The lefthand graph shows the crosssection of the right cylinder bounding the region. The (vertical) orange arrow indicates the direction of the inner integral. The righthand graph shows the volume swept by the integral as set up via the dataentry fields at the top of the template.
The heading on the righthand graph is quoted because not every double integral is in reality a volume. True enough, the iterated double integral can be interpreted as a volume, but every such integral need not necessarily be a volume calculation.
The faces of the threedimensional region in the righthand graph are color coded as per the schematic
with blue used for any facet in the plane and red used for the surface defined by the integrand .


Visualizing Regions of Integration in 3D Cartesian Coordinates


Evaluate and Graph

Volume Element

, where



To compute a volume, use . We have selected the volume element to be , a choice that simplifies the integration. Had we selected , for example, it would have taken two triple integrals to determine the same volume.
The faces of the threedimensional region in the graph are colorcoded as per the schematic
In this example, the yellow and gray facets do not appear because of the way the green and brown facets join.


Visualizing Regions of Integration in Polar Coordinates


The lefthand graph is an animation showing how the radius "vector" sweeps the region . The righthand graph interprets the integral as the calculation of a volume.


Visualizing Regions of Integration in Cylindrical Coordinates


Use cylindrical coordinates to calculate the volume of the solid cut from a unit sphere by a cone whose vertex is at the center of the sphere, and whose generator makes a angle with the axis of symmetry of the cone.

Evaluate and Graph


, where





Visualizing Regions of Integration in Spherical Coordinates


Use spherical coordinates to calculate the volume of the solid cut from a unit sphere by a cone whose vertex is at the center of the sphere, and whose generator makes a angle with the axis of symmetry of the cone.

This same volume was computed using cylindrical coordinates in the previous section.
Evaluate and Graph


, where




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