<A HREF="https://arxiv.org/abs/1410.1212">New Approximations for the Area of the Mandelbrot Set</A> gives the state of the art from 2014, and a related paper from 2015 is <A HREF="https://link.springer.com/article/10.1007/s11071-015-1917-4">On a numerical approximation of the boundary structure and of the area of the Mandelbrot set</A>. An area of 1.5052 up to five significant digits is one estimate, an upper bound is 1.68288, a lower bound is 1.3744.