Integral Calculus: JEE Main 2026 Mathematics question

Let $e$ be the base of natural logarithm and let $f:\{1,2,3,4\}\to\{1,e,e^2,e^3\}$ and $g:\{1,e,e^2,e^3\}\to\left\{1,\frac12,\frac13,\frac14\right\}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $\phi(x)=\left[f^{-1}\left\{g^{-1}\left(\frac12\right)\right\}\right]^x$, then the area of the region $R=\{(x,y):x^2\le y\le\phi(x),\ 0\le x\le1\}$ is: