Answer
$$2\ln 2 - 3$$
Work Step by Step
$$\eqalign{
& \int_0^{\ln 2} {\int_{{e^x}}^2 {dydx} } \cr
& {\text{Integrate with respect to }}y \cr
& = \int_0^{\ln 2} {\left[ y \right]_{{e^x}}^2dx} \cr
& = \int_0^{\ln 2} {\left( {2 - {e^x}} \right)dx} \cr
& {\text{Integrate }} \cr
& = \left[ {2x - {e^x}} \right]_0^{\ln 2} \cr
& {\text{Evaluating}} \cr
& = \left[ {2\ln 2 - {e^{\ln 2}}} \right] - \left[ {2\left( 0 \right) - {e^0}} \right] \cr
& = 2\ln 2 - 2 - 1 \cr
& = 2\ln 2 - 3 \cr} $$
