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i would have to posit that sources act like cycles. One thing leads to another, and another and ultimately winds up creating something that leads to itself. These things can be extraordinarily complex, such as the cycle of the magnetic/electric field: increasing B decreases E at an increasing rate of decay, the increasing rate works to increase the Rate of increasing B while B decreases due to decreasing E. The citric acid cycle, etc. Now, how do the cycles appear? Well, that in itself is a question I'm not willing to think about quite yet.
"You cannot find the area under the curve e^(x^2) dx in terms of a simple elementary function. "
The area under the curve e^x^2 equals [SUM(FROM_n=0_TO_n=INFINITY) OF {(x^(2n+1))/((2n+1)(n!))} MINUS SUM(FROM_n=0_TO_n=INFINITY) OF {(a^(2n+1))/((2n+1)(n!))}] for all x greeater than or equal to a. This equation, derived from the formal definition of e (via a taylor series expansion), is impossible to evaluate predcisely because you have to keep adding terms forever. In practice, a finite limit on n is necessary. You can get good results by expressing the summation recursively, and then forcing the ratio of the n+1 coefficient over the n coefficient equal to zero for some n.
Hope this helps. I apologize in advance if I did not successfully adress the topic of your question. I am especially hesitant to talk as though I know the solution because I am uncertain to what a simple elementary function is. Could you give an example of a simple function that is not elementary, and an elementary function that is not simple? I should look these things up before posting, yes, yes, but the examples would help. If you were to try to apply the aforementioned equation in your java script application, via recursive relation formulas, I'm sure you'd get some good results.
Also ..
For those who aren't familiar with the theorem of Pappus, pappus discovered that the volume of a solid of revolution is equal to the area of the region, multiplied by the distance the centroid travels as the region is revolved. (centroid meaning the center of gravity) if we revolve the region y= e^(x^2) about the y axis, this volume (which we can find) is equal to the area of the region (which we don't know) times the x coordinate of the regions center of gravity, or centroid. (which we also can't find.)
If a=-x, then the centroid of the x-coordinate is 0. Otherwise, the x-coordinate, as you say, cannot be determined; however, I would venture to say the x-coordinate centroid equals positive infinity (domain of integration: FROM_0_TO_POSITIVE.INFINITY along the x-axis), because you must at least be certain that the x-coordinate centroid is not finite, for if you were to pick a point that you thought was a finite x-coordinate centroid, then I'd elongate the domain of 0 to n (along the x-axis) even farther, ever so far as to continue to push the x-centroid into an infinite region.
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