The area of the lune is the area of the upper half of the small circle, plus the area of the triangle formed by the two radii of the large circle and the diameter of the small circle (see the attachment), minus the area of the sector of the large circle bounded by those two radii. So the area of the lune is
$\displaystyle \boxed{\tfrac12\pi r^2 + r\sqrt{R^2-r^2} - R^2\arcsin(r/R)}$.
(Seeing this problem in the Calculus section, I started trying to calculate the area as a double integral using polar coordinates. It's possible to do it that way, but the geometric approach is far easier!)