Answer
$$A = \frac{9}{2} - \frac{1}{2}{e^{ - 4}}$$
Work Step by Step
$$\eqalign{
& {\text{From the graph we can see that the region is given by:}} \cr
& A = \int_0^2 {\left( {{e^{ - 2x}} + 2} \right)} dx \cr
& {\text{Integrate and evaluate}} \cr
& A = \left[ { - \frac{1}{2}{e^{ - 2x}} + 2x} \right]_0^2 \cr
& A = \left[ { - \frac{1}{2}{e^{ - 2\left( 2 \right)}} + 2\left( 2 \right)} \right] - \left[ { - \frac{1}{2}{e^{ - 2\left( 0 \right)}} + 2\left( 0 \right)} \right] \cr
& A = - \frac{1}{2}{e^{ - 4}} + 4 + \frac{1}{2} \cr
& A = \frac{9}{2} - \frac{1}{2}{e^{ - 4}} \cr
& A \approx 4.4908 \cr} $$
