In the picture, you want the area of the region CXDY. Split it into two by the line CD. The part to the right of that line is equal to the sector ACXD of the smaller circle, less the triangle ACD. Similarly, the part to the left of the line is equal to the sector BCYD of the larger circle, less the triangle BCD.

You will doubtless have noticed that ACB and ADB are 3-4-5 right-angled triangles, which will be helpful in calculating the angles that you require in order to find the areas of those sectors and triangles.