In this article we seek finding its surface area.
We can calculate the area inside of the curve by using the Green's theorem from potential theory, that is basically the same as the Stokes theorem applied to 2D.
It's clear also that , but since we already showed that the whole set becomes the unit disk in the limit, we can conclude that the relative area of the rest of hyperbolic components vanish as the degree of the polynomial increases. This is something that agrees with the graphical experiments.
Much has been discussed about the area of the standard Mandelbrot set. Only one mathematically exact solution has been proposed, but it happens to be far too slow to use in any computer. The area is given as a summation series, may be a bit disappointing for many people that would expect a nice small and compact formula, probably involving a few universal constants ( pi, Feigenbaun number, the golden ratio, e, ...). There has been many experiments in other directions, though, as pixel counting approaches (sometimes assisted by the distance estimator to better bound the error).
Of course the whole area approaches as k grows, just as we predicted theoretically.