Let e be the base of natural logarithm and let f:{1,2,3,4} →{1, e, e^2, e^3} and g:{1, e, e^2, e^3} →{1, 1/2, 1/3, 1/4} be two bijective functions…

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