Answer
$$A = - \frac{1}{2}\left( {{e^{ - 6}} - {e^2}} \right)$$
Work Step by Step
$$\eqalign{
& {\text{From the graph we can see that the region is given by:}} \cr
& A = \int_{ - 1}^3 {{e^{ - 2x}}} dx \cr
& {\text{Integrate and evaluate}} \cr
& A = \left[ { - \frac{1}{2}{e^{ - 2x}}} \right]_{ - 1}^3 \cr
& A = - \frac{1}{2}\left[ {{e^{ - 2x}}} \right]_{ - 1}^3 \cr
& A = - \frac{1}{2}\left[ {{e^{ - 2\left( 3 \right)}} - {e^{ - 2\left( { - 1} \right)}}} \right] \cr
& A = - \frac{1}{2}\left( {{e^{ - 6}} - {e^2}} \right) \cr
& A \approx 3.69328 \cr} $$