Classroom Tips and Techniques: Visualizing Regions of Integration

Robert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow
Maplesoft 

Introduction 

 

My favorite example for teaching students how to set up triple integrals over a specified three-dimensional region was the calculation of the first-octant volume bounded by the cylinder and the plane .  I computed this volume in Cartesian coordinates using all six possible orders of integration, and for visualization, used code written by Tim Murdoch who taught at Washington and Lee University in Lexington, VA, in the early 1990s.  His code (calcplot) was apparently inspired by the mvcal code that accompanied a text by Cheung and Harer. 

 

Commands in the calcplot package would draw a region of integration using the limits of integration that would integrate over the region.  The visual feedback was an essential ingredient in learning how to formulate iterated integrals.  Careful inspection of the region in my favorite example shows it is the scrap cut off from a piece of quarter-round molding when an end is mitered for an inside corner.  In fact, being an amateur woodworker, I cut a bagful of such scraps so my students could each manipulate the solid object whose volume we were calculating six different ways. 

 

Over the years, I updated the calcplot code and continued using it with my students, hoping that one day, Maple would build in a similar functionality.  In this month's article, the synergy between the visual and the analytic is demonstrated  with a learning tool built with Maple's embedded components. 

 

Initializations 

 

 

 

Preliminary Graphics 

 

Figure 1 shows the first-octant intersection of the cylinder and the plane . 

 

Figure 1   Intersection of the cylinder and the plane

 

The portion "in front of" the plane is the volume to be computed.  This computation requires that we know the curve of intersection of the plane and the cylinder.  This curve is visualized in Figure 2. 

 

Figure 2   The curve of intersection

 

In particular, it is essential to know the analytic representation of the projection of this curve onto the -plane and the -plane.  Rotating the graph in Figure 2 suggests that these projections are parts of circles.  The projection onto the -plane is clearly a circular arc from since it must align with the cylinder itself.  The projection onto the -plane is obtained by eliminating from the equations . One readily obtains . 

 

Figure 3, showing the region of integration as a solid, is drawn with code captured in our learning tool built with embedded components.  In essence, each of the six possible bounding faces is drawn parametrically, and joined together in a single plot. 

 

Figure 3   The region of integration as a solid object

 

The Six Iterated Integrals 

 

For the region in Figure 3, Table 1 lists the six possible iterated integrals that give the volume in Cartesian coordinates.  Each integral evaluates to . 

 

Table 1   Six iterated integrals for the volume of the region in Figure 3

 

Because of the way we typically graph in three dimensions, it would seem that the simplest integration orders to visualize begin with .  However, because the axis of the cylinder aligns with the -axis, the integration orders starting with are surprisingly simple in this example.  The most surprising of the six iterations is the one with order where the two innermost integrals have the same upper limit of integration! 

 

Planar Regions 

 

There are just two orders of integration for an iterated double integral over a planar region.  Some calculus texts distinguish between type 1 and 2 regions, depending on whether the region is amenable to the integration order or , respectively.  Some regions, such as the one bounded by the curves and , are of both types simultaneously, as shown in Figure 4. 

 

Figure 4   Planar region bounded by and

 

One would think that the simplicity of iteration with the region in Figure 4 would be preserved for the solid bounded by the surfaces .  Our second example will show that this is not the case. 

 

Example 2 

 

Figure 5 shows the solid whose bounding surfaces are given by . 

 

Figure 5   The solid bounded by

 

Its volume is , as determined by either of the iterated integrals in Table 2. 

 

Table 2   Volume of solid in Figure 5 computed by iterations starting with

 

Any of the other four iterated integrals for this volume require splitting the region into two parts, as per the calculations listed in Table 3. 

 

Table 3   Iterated integrals for the volume of the solid in Figure 5

 

Writing the integrals in Table 3 requires knowing the intersections of the surface with the surfaces and . These space curves are shown in Figure 6, the first curve in red, the second in green. 

 

Figure 6   Intersection of with in red, and with in green

 

Figures 7 and 8 project the curves in Figure 6 onto the -plane and -plane, respectively. 

 

Figure 7   Projection of Figure 6 onto the -plane	Figure 8   Projection of Figure 6 onto the -plane

 

Note that in the -plane, the red curve () lies above the green (), and in the -plane, the orientation is reversed.  This is easier to understand if the analytic representations of these curves and their projections are obtained.  The red curve in Figure 6 is given parametrically by , whereas the green one is given by . 

 

The projection of the red curve onto the -plane is given by ; the green, by .  The projection of the red curve onto the -plane is given by ; the green, by . 

 

It should now be possible to write the integrals in Table 3, a task make easier by the use of the learning tool given in the next section. 

 

Learning Tool for Regions of Integration 

 

Evaluate and Graph
Volume Element	, where

 

After an order of integration is selected, the data entry is obvious.  An integrand, and the limits of integration are entered.  The "Exact Value" button computes the resulting integral in closed form, whereas the "Floating-Point Value" button computes numerically.  The "Plot" button draws a graph of the region corresponding to the given limits of integration. 

 

The six surfaces (possibly fewer) in the graph of the region of integration are color-coded.  The surface from which the inner integration begins is drawn in blue; where it ends is drawn in red.  The surface from which the middle integration begins is drawn in green; where it ends is drawn in gold.  The surface from which the outer integration begins is drawn in yellow; where it ends is drawn in gray. 

 

Similar learning tools have been created for Cartesian coordinates in the plane, for polar coordinates, cylindrical coordinates, and spherical coordinates. 

 

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