Unusual integral notation When I was learning analysis, I often wondered why I couldn't seem to find anything like $$\iint f(x) (dx)^2$$ in a standard calculus text, and concluded that it should be meaningless – even though, since we can differentiate functions multiple times, it would make sense that we can also integrate them repeatedly.
But then I stumbled upon this blog entry by the creator of Mathematica, showing that Leibniz had similar notation in mind when he was developing the calculus, and found out about the differintegral operator, using which the above expression looks like $D^{-2}[f(x)]$.
My question is, why don't we see this notation that often in basic analysis courses? What is the graphical meaning of such an expression – i.e. how would its behavior affect the shape of $f(x)$? And how would one solve it? Is there even a definite analog of it, and if so what is its geometrical meaning?
 A: like @Mathaholic said, if you write out the operations involved in integrating a function twice, you get:
$$\int \left(\int f(x) dx\right) dx$$
It looks like this has been compressed via the following process:
$$ \int \left(\int f(x) dx\right) dx \to \int \int f(x) dx dx \to \int \int f(x) (dx)^2$$
It's a little confusing, since it suggests to me we are integrating a single function over the Cartesian product $x\times x$. But $f$only has one argument.
Basically, its mixing geometric and operator notation, much like $\frac{d^2}{dx^2}$ is defined more inline with operator theory. I've seen this quite a bit in time series analysis, where we use the Backshift $(B)$ operator as a variable: 
$$(1-B)Z_t = Z_t - BZ_t = Z_t-Z_{t-1}$$
A key theoretical issue in time series is finding "unit roots", where you solve expressions like:
$$(B^2+2B+1) = 0 $$
To find if the time series:
$$(B^2+2B+1)Z_t$$
Is stationary.
This is a bit of a diversion, but I was reminded by this example of how operator notation can be very confusing if you're not used to to.
