Area Bounded by Exponential Region and Parabola

To find the area of the given region, we first determine the explicit form of the function $\phi(x)$.

Step 1: Determine the function $f(x)$ The function $f : \{1, 2, 3, 4\} \to \{1, e, e^2, e^3\}$ is bijective and strictly decreasing. Arranging the elements of the domain and codomain in increasing order:

• Domain: $1 < 2 < 3 < 4$
• Codomain: $1 < e < e^2 < e^3$

Since $f$ is strictly decreasing: $f(1) = e^3$ $f(2) = e^2$ $f(3) = e$ $f(4) = 1$

Thus, $f^{-1}(e^2) = 2$.

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Step 2: Determine the function $g(x)$ The function $g : \{1, e, e^2, e^3\} \to \left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right\}$ is bijective and strictly increasing. Arranging the elements of the codomain in increasing order: $\frac{1}{4} < \frac{1}{3} < \frac{1}{2} < 1$

Since $g$ is strictly increasing: $g(1) = \frac{1}{4}$ $g(e) = \frac{1}{3}$ $g(e^2) = \frac{1}{2}$ $g(e^3) = 1$

Thus, $g^{-1}\left(\frac{1}{2}\right) = e^2$.

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Step 3: Evaluate $\phi(x)$ Substituting $g^{-1}\left(\frac{1}{2}\right) = e^2$ into the expression for $\phi(x)$: $\phi(x) = \left[ f^{-1} \left\{ g^{-1} \left( \frac{1}{2} \right) \right\} \right]^x = \left[ f^{-1}(e^2) \right]^x = 2^x$

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Step 4: Calculate the Area of the Region $R$ The region $R$ is defined by: $R = \{(x, y) : x^2 \le y \le 2^x, 0 \le x \le 1\}$

Since $2^x \ge x^2$ for all $x \in [0, 1]$, the area $A$ is given by the definite integral: $A = \int_{0}^{1} \left( 2^x - x^2 \right) dx$

Evaluating the integral step-by-step: $A = \left[ \frac{2^x}{\log_e (2)} - \frac{x^3}{3} \right]_{0}^{1}$

$A = \left( \frac{2^1}{\log_e (2)} - \frac{1^3}{3} \right) - \left( \frac{2^0}{\log_e (2)} - \frac{0^3}{3} \right)$

$A = \frac{2}{\log_e (2)} - \frac{1}{3} - \frac{1}{\log_e (2)}$

$A = \frac{1}{\log_e (2)} - \frac{1}{3} = \frac{3 - \log_e (2)}{3 \log_e (2)}$

Thus, the area of the region is $\frac{3 - \log_e (2)}{3 \log_e (2)}$.