This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet 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 email@example.com 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. Jay Hill's latest results using Root Solving and Component
Series Evaluation shows the area is at least 1.506302 and less than
1.5613027. See Fractal Horizons edited by Cliff Pickover and Hill's
home page for details about his work.
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
World Wide Web (in Postscript format)
3. Jay Hill's Home page which includes his latest updates.
Jay's Hill Home Page via the World Wide Web.