Using the shell method, the volume is
[tex]V=\displaystyle2\pi\int_0^3x\cdot5xe^{-x}\,\mathrm dx=10\pi\int_0^3x^2e^{-x}\,\mathrm dx[/tex]
Integrate by parts, taking
[tex]u=x^2\implies\mathrm du=2x\,\mathrm dx[/tex]
[tex]\mathrm dv=e^{-x}\,\mathrm dx\implies v=-e^{-x}[/tex]
[tex]V=\displaystyle10\pi\left(-x^2e^{-x}\bigg|_0^3+2\int_0^3xe^{-x}\,\mathrm dx\right)[/tex]
[tex]V=\displaystyle-90\pi e^{-3}+20\pi\int_0^3xe^{-x}\,\mathrm dx[/tex]
Integrate by parts again, with
[tex]u=x\implies\mathrm du=\mathrm dx[/tex]
[tex]\mathrm dv=e^{-x}\,\mathrm dx\implies v=-e^{-x}[/tex]
[tex]V=-90\pi e^{-3}+20\pi\left(-xe^{-x}\bigg|_0^3\displaystyle\int_0^3e^{-x}\,\mathrm dx\right)[/tex]
[tex]V=-150\pi e^{-3}+20\pi\displaystyle\int_0^3e^{-x}\,\mathrm dx[/tex]
[tex]V=\displaystyle-150\pi e^{-3}+20\pi(-e^{-x})\bigg|_0^3[/tex]
[tex]\boxed{V=20\pi-170\pi e^{-3}}[/tex]
