Discussion:  All Activities in Calculus on Computer 
Topic:  Intigration 
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Subject:  RE: Intigration 
Author:  Alan Cooper 
Date:  May 5 2006 
> On May 4 2006, Mathman wrote:
>> On May 3 2006, ihor wrote:
>>> ...Should they be available at all times?...
>> Sigh. I knew this might be a can of worms ...
> We have been down this road a few times ...
I agree that we have gone off topic (having started with issues around need,
use, and availability of tools for symbolic integration, and ending up with
appropriateness of technology in all grade levels and "at all times") but I am
not sure who is responsible and I apologize if I have contributed to the
diversion.
What I am sure of is that our diversion in this instance is not exclusively
David's responsibility, and that, as a developer and user of Math Tools myself,
I appreciate hearing skeptical voices asking me how, or if at all, the fruits of
my labour really do improve the learning experience. I will try to address the
creation and possible utility of a dynamic geometry illustration of the
construction that I presume David has in mind  either in a new thread or back
where he originally posted it (if I can find it), but that may take a while and
in the meantime I'd like to return to symbolic integration.
The example we started with *is* a good one, because the integrals that arise in
applied problems (especially but not exclusively eg in time value of money
problems, in electric discharge problems, and in the use of Laplace transforms
to solve differential equations) often involve combinations of exponential and
algebraic functions for which there is no closed form answer in terms of
elementary functions. The availability of symbolic and numerical integration
tools allows for the assignment of much more realistic applied problems to
students than was posssible in the past  and time not spent on the classical
"techniques of integration" becomes available to explore other more interesting
mathematics as well as the more realistic applications.
But those classical techniques can sometimes be used to prove important
relationships between different quantities (or even to get exact answers in
special cases), and some attention to them may still be useful for helping to
develop an understanding of these deeper relationships. So the question of how
far they should be deemphasized remains a tricky one.
Alan
 
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