Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.4 Graphs And Applications Involving Logarithmic And Exponential Functions - Exercises Set 6.4 - Page 440: 41

$$A = \frac{3}{2}$$

$$\eqalign{ & {\text{Let }}f\left( x \right) = {e^{2x}};{\text{ }}\left[ {0,\ln 2} \right] \cr & {\text{The area is given by}} \cr & A = \int_0^{\ln 2} {{e^{2x}}} dx \cr & {\text{Integrating}} \cr & A = \left[ {\frac{1}{2}{e^{2x}}} \right]_0^{\ln 2} \cr & A = \frac{1}{2}\left[ {{e^{2x}}} \right]_0^{\ln 2} \cr & A = \frac{1}{2}\left[ {{e^{2\left( {\ln 2} \right)}} - {e^{2\left( 0 \right)}}} \right] \cr & {\text{Simplifying}} \cr & A = \frac{1}{2}\left[ {4 - 1} \right] \cr & A = \frac{3}{2} \cr} $$