Find the area of the region inside the circle (x-1)^2+y^2=1 and outside the circle x^2+y^2=1

There are numerous ways to approach this. Perhaps the simplest is to subtract twice the area of the shaded segment (see the attached figure) from that of the circle.

The area of the shaded segment is
  segment area = (1/2)r^2*(θ -sin(θ))
where r=1 and θ=2π/3

The area of the circle, of course, is πr^2 = π, so the area of interest is ...
  A = π -(2π/3 -sin(2π/3))
  A = π/3 +(√3)/2