Answer
$$V = 2\pi \left( {1 - 4{e^{ - 3}}} \right)$$
Work Step by Step
$$\eqalign{
& y = {e^{ - x}},{\text{ }}y = 0,{\text{ }}x = 0,{\text{ }}x = 3 \cr
& {\text{Calculate the Volume by cylindrical shells about the }}y{\text{ - axis}} \cr
& V = 2\pi \int_a^b {xf\left( x \right)} dx,{\text{ then}} \cr
& {\text{The Volume of the solid is given by}} \cr
& V = 2\pi \int_0^3 {x{e^{ - x}}} dx \cr
& {\text{Integrating by parts}} \cr
& u = x,{\text{ }}dv = {e^{ - x}}dx \cr
& du = dx{\text{ }}v = - {e^{ - x}} \cr
& \int {x{e^{ - x}}dx = - x{e^{ - x}} - \int {\left( { - {e^{ - x}}} \right)dx} } \cr
& {\text{ }} = - x{e^{ - x}} + \int {{e^{ - x}}dx} \cr
& {\text{ }} = - x{e^{ - x}} - {e^{ - x}} \cr
& V = 2\pi \int_0^3 {x{e^{ - x}}} dx = - 2\pi \left[ {x{e^{ - x}} + {e^{ - x}}} \right]_0^3 \cr
& V = - 2\pi \left[ {3{e^{ - 3}} + {e^{ - 3}}} \right] + 2\pi \left[ {0{e^0} + {e^{ - 0}}} \right] \cr
& {\text{Simplifying}} \cr
& V = - 2\pi \left[ {4{e^{ - 3}}} \right] + 2\pi \left[ 1 \right] \cr
& V = 2\pi - 8\pi {e^{ - 3}} \cr
& V = 2\pi \left( {1 - 4{e^{ - 3}}} \right) \cr} $$
