Answer
$$m = \frac{{5\left( {1 - {e^{ - 8}}} \right)}}{2}$$
Work Step by Step
$$\eqalign{
& \rho \left( x \right) = 5{e^{ - 2x}},{\text{ }}0 \leqslant x \leqslant 4 \cr
& {\text{The mass of the object is }}m = \int_a^b {\rho \left( x \right)} dx,{\text{ }}\left( {{\text{See page 460}}} \right) \cr
& m = \int_0^4 {5{e^{ - 2x}}} dx \cr
& {\text{Integrating}} \cr
& m = \left[ { - \frac{5}{2}{e^{ - 2x}}} \right]_0^4 \cr
& m = - \frac{5}{2}\left[ {{e^{ - 2\left( 4 \right)}} - {e^{ - 2\left( 0 \right)}}} \right] \cr
& m = - \frac{5}{2}\left( {{e^{ - 8}} - 1} \right) \cr
& m = \frac{{5\left( {1 - {e^{ - 8}}} \right)}}{2} \cr} $$
