Let \(e\) be the base of natural logarithm and let \(f:{1,2,3,4}\to \left{1,e,{e}^{2},{e}^{3}\right}\) and \(g:\left{1,e…

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

[JEE Main 2026, 6 Apr (Shift 1)]