Question

Find the area A of the crescent-shaped shaded region (called a lune) bounded by arcs of circles with radii r and R. (See the figure below. Assume the center of the smaller circle is collinear with the two points of intersection.)!

Answer 1

Consider the following graph The equation of the large circle is x'+y R The equation of the small circle is x2(y-b)-r This can be written as y-b+Vr'-x', Where b- Ris the distance between the centers of the circles Therefore, the desired area is, Take first integral 21 bax = 2br Second integral Let also cos θ Therefore, the integral becomes: -1 | X Therefore, from equation (1) = 2rVR'-r2-r2 sin-1 (1)-| R2 sin-1 Hence, the area of the shaded region is-r R-2+-r2-R, sin-i