Why is the median of a PDF not always equal to the mean? When studying probability density functions(PDFs), the focus is often placed on the "mean" and not the "median". The mean is often described as the center of mass of the PDF. I've always taken this as true.

However, the definition of the median seems to correspond to what I would consider the center of mass. That definition being: 

$\int_{-\infty}^{median}pdf(x)dx = \frac{1}{2}$

Looking at this, we're essentially integrating the PDF until we reach the half-way point. To me, this seems identical to what we would consider the center of mass.

So why does integrating a PDF until we reach halfway (the median) not always equal the mean if the mean is the center of mass?

Note that I understand the difference between the mean and median from a general statistics perspective, but I'm curious from a PDF perspective.

I'll also format the question in the following manner:

$\int_{-\infty}^{\infty}pdf(x)dx = mass$

$\int_{-\infty}^{\infty}x*pdf(x)dx = center\:of\:mass$

If we define the following equation:

$\int_{-\infty}^{T}pdf(x)dx = \frac{mass}{2}$

Why is T not always the center of mass? It appears T is the center of mass if the function is symmetric, but why doesn't it hold true if the function is non-symmetric?
If you consider the pdf as defining the distribution of mass in a thin rod of infinite length and variable density (and so the "weight per inch" of the rod is different depending on which part of the rod we are talking about), then the mean is the point about which the rod is perfectly balanced (that is, if we set up a fulcrum and place the rod on it, it will be horizontal and balanced perfectly if the fulcrum is at the mean, and will tilt downwards on one side or the other if the fulcrum is anywhere else.

The median, on the other hand, is the point for which we can say that half the mass is to the left of the median and half the mass is to the right of the median. If that is what floats your boat as the one true and the only meaning of "center of mass" and you feel that everyone else who believes otherwise is in a state of mortal sin, so be it. Just be aware that when you refer to the median as the center of mass, you are likely to be misunderstood by those lesser mortals who have not attained your state of enlightenment and who still believe that center of mass is another name for what they call the mean of the pdf or random variable.

With respect to the comments on the main question as to an asymmetric pdf means that the mean and median will be different, this answer over on stats.SE (where this question more properly belongs) gives an example of an asymmetric pdf for which the mean, median and mode all have the same value.