I was thinking about the area problem, yet again, and found myself asking the question: why must we go through such elaborate means to get Maple to generate a plot of the region between two (or more curves)? I use the word elaborate to describe any process that would might become overwhelming, for, say a student, to go through to accomplish a task. Anyone with the most basic of backgrounds can understand the area problem, but yet, such an individual might not find it a trivial matter to have Maple produce such a region.
I (naturally or unnaturally) started thinking about volumes and how much better Maple is at generating solids of revolutions. Looking more closely at the VolumeOfRevolution( ) command (in the Student[Calculus1] package), there is the output=region option, that simply shows the two-dimensional region, which when revolved about some vertical or horizontal axis, yields the solid. So Maple actually does have a not-so-elaborate way of generating these regions. So I tried an example, which in terms of x, would require the sum of two integrals (either an area or a volume calculation). This example is attached. I was very pleased with the result.
So what is the point of this post? Well, I would like the engineers at Maplesoft to consider using whatever procedure they have that facilitates the output=region option to use this procedure to create a new command, say, AreaBetweenCurves, with obvious syntax, that would find its way into the wonderfully powerful tools in the Student[Calculus1] package. i may be off in thinking that current way generating such regions is elaborate, and maybe I was the last person to figure out it could be done this way. Nonetheless, I needed to share my discovery with this community, and, in my little way, further the discussion on this topic.2D_Region_Using_Volu.mw