Area Under The Curves

Let $e$ be the base of natural logarithm and let $f:\{1,2,3,4\} \rightarrow\left\{1, e, e^2, e^3\right\}$ and $\mathrm{g}:\left\{1, e, e^2, e^3\right\} \rightarrow\left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right\}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $\phi(x)=\left[f^{-1}\left\{g^{-1}\left(\frac{1}{2}\right)\right\}\right]^x$, then the area of the region $\mathrm{R}=\left\{(x, y): x^2 \leq y \leq \phi(x), 0 \leq x \leq 1\right\}$ is :