Let \( e \) be the base of natural logarithm and let \(\alpha/\beta\) and \( g : \{1 \), \( e \), \( e^2 \), \( e^3 \} \to \{ 1 \), \( \tfrac{1}{2} \), \( \tfrac{1}{3} \), \( \tfrac{1}{4} \} \) be two bijective functions such that \( f \) is strictly decreasing and \( g \) is strictly increasing. If \( \phi(x)=\left[ f^{-1}\!\left\{\,g^{-1}\!\left( \tfrac{1}{2} \right)\!\right\}\right]^x \), then the area of the region \( R=\{(x,y): x^2 \le y \le \phi(x),\; 0 \le x \le 1\} \) is:
