Graviational Field of Floating Mountain

Homework Statement

Consider a spherical mountain of radius a, mass M, floating in equililbrium in the earth, and whose density is half that of the earth. Assume that a is much less than the earth's radius, so that the earth's surface can be regarded as flat in the neighborhood of the mountain. If the mountain were not present, the graviational field intensity near the earths's surface would be g0. Find the difference between g0 and the actual value of g at the the top of the mountain.

The Attempt at a Solution

Knowledge of Lagrangian Mechanics is not required. I determined the mountain is half submerged below the earth's surface and g0 is independent of height (infinite disk). I attempted solving for the g field of a hemisphere but the intergral was not a form that could be solved without binomial expansion. The answer provided in the back of the text is

g0 - g = (GM/a^2)(2 - sqrt(2))

I noted the answer is equivalent to a thin disk of radius a, mass M, with surface density equal to the earth's volume density at a distance a from the center of the disk along the disk axis. So, somehow superposition shows the equivalence. Any hints are appreciated.