Answer
$$A = - 2\left[ {{e^{ - 3/2}} - 1} \right]$$
Work Step by Step
$$\eqalign{
& {\text{From the graph we can see that the region is given by:}} \cr
& A = \int_0^{\sqrt 6 } {x{e^{ - {x^2}/4}}} dx \cr
& {\text{Integrate and evaluate}} \cr
& A = - 2\int_0^{\sqrt 6 } {\left( { - \frac{1}{2}x} \right){e^{ - {x^2}/4}}} dx \cr
& A = - 2\left[ {{e^{ - {x^2}/4}}} \right]_0^{\sqrt 6 } \cr
& A = - 2\left[ {{e^{ - {{\left( {\sqrt 6 } \right)}^2}/4}} - {e^{ - {{\left( 0 \right)}^2}/4}}} \right] \cr
& A = - 2\left[ {{e^{ - 3/2}} - {e^0}} \right] \cr
& A = - 2\left[ {{e^{ - 3/2}} - 1} \right] \cr
& A \approx 1.5537 \cr} $$