This problem requires a strong understanding of conic sections (hyperbola and parabola) and the application of definite integration to calculate the area of a region. We will break down the solution into several key steps: first, analyzing the hyperbola to find its axes; second, analyzing the parabola; third, sketching the region and determining the limits of integration; and finally, evaluating the definite integral.

1. Key Concepts and Formulas

The core concept here is finding the area between two curves using definite integration. If a region is bounded by $y=f(x)$ from above and $y=g(x)$ from below, between $x=a$ and $x=b$, its area is given by: