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**Fractional Calculus...? What??**

I came across this Wiki article a couple of days ago:

http://en.wikipedia.org/wiki/Fractional_calculus

As a student who just finished an undergrad major in math, the idea of a "fractional derivative" or "fractional integral" is mind blowing! Up until I read the article, I pretty much thought that the differential and integral operators were "fixed."

Let f:R -> R be a fcn.

Geometrically speaking, if the derivative of a fcn f at a point p in its domain is the rate of change of the fcn at that point and the integral of a fcn over an interval in the domain is the "area under the curve," how can I interpret the fractional derivative and fractional integral?