The problem lies in finding the small circle area; since its radius doesn't start from the square corner, we cannot just subtract the Bigger circle radius from the square diameter and divide by 2.
The solution i came with is finding the ratio between (the the bigger circle radius) and (the part of the diameter that is crossing over the blue shaded area + circle radius)...
.
This will let us know how to calculate the small circle radius by knowing only the length of the line that is crossing over the blue shaded area.
.
So the shaded section area (inlc small circle) :
(10*10pi*5^2)/4)

small circle area:
pi*(((5sqrt(2)5)(5sqrt(2)5)*(5sqrt(2)5)/(10+(5sqrt(2)5)))/2)^2
=
3.053
.
This is the calculation I came up with:
http://bit.ly/2mbAFqt