Answer
$A = \int_0^1 {\left( {{3^x} - {2^x}} \right)} dx$
Work Step by Step
$$\eqalign{
& {\text{Let }}y = {2^x}{\text{ and }}y = {3^x},{\text{ }}x = 1 \cr
& {\text{Find the intersection points between }}y = {2^x}{\text{ and }}y = {3^x} \cr
& y = y \cr
& {2^x} = {3^x} \cr
& {\text{Solving}} \cr
& x = 0 \cr
& {\text{The enclosed area is shown in the graph below}}{\text{.}} \cr
& {3^x} \geqslant {2^x}{\text{ on the interval }}\left[ {0,1} \right] \cr
& {\text{Therefore}} \cr
& {\text{The area is given by}} \cr
& A = \int_0^1 {\left( {{3^x} - {2^x}} \right)} dx \cr} $$
