1. Calculate angle CBA using the sine rule: \(\angle CBA = \sin^{-1}\left(\frac{7}{8}\right) \approx 1.0654 \text{ radians}\).

2. Calculate the area of sector BCYD: \(\frac{1}{2} \times 8^2 \times 2 \times 1.0654 \approx 68.19 \text{ cm}^2\).

3. Calculate the area of triangle BCD: \(7 \times \sqrt{8^2 - 7^2} \approx 27.11 \text{ cm}^2\).

4. Calculate the area of the semicircle CXD: \(\frac{1}{2} \pi \times 7^2 \approx 76.97 \text{ cm}^2\).

5. Total area enclosed by the arcs: \(68.19 - 27.11 + 76.97 \approx 118.0 \text{ to } 118.1 \text{ cm}^2\).