This article is from the Fractal FAQ, by Ermel Stepp email@example.com with numerous contributions by others.
Ewing and Schober computed an area estimate using 240,000 terms of the
Laurent series. The result is 1.7274... However, the Laurent series
converges very slowly, so this is a poor estimate. A project to measure the
area via counting pixels on a very dense grid shows an area around 1.5066.
(Contact firstname.lastname@example.org for more information.) Hill and Fisher used
distance estimation techniques to rigorously bound the area and found
the area is between 1.503 and 1.5701.
1. J. H. Ewing and G. Schober, The Area of the Mandelbrot Set, _Numer.
Math._ 61 (1992), pp. 59-72.
2. Y. Fisher and J. Hill, Bounding the Area of the Mandelbrot Set,
_Numerische Mathematik_, . (Submitted for publication). Available by
ftp: legendre.ucsd.edu:/pub/Research/Fischer/area.ps.Z ..