Answer
$\displaystyle{V=2\pi e}$
Work Step by Step
$\displaystyle{V=\int_{-1}^0 \left(2\pi (1-x)\right)\left(e^{-x}\right)\ dx}\\
\displaystyle{V=2\pi\int_{-1}^0 e^{-x}-xe^{-x}\ dx}\\
\displaystyle{V=2\pi\int_{-1}^0 e^{-x}\ dx-2\pi\int_{-1}^0xe^{-x}\ dx}\\$
$\displaystyle \left[\begin{array}{ll} u=x & dv=e^{-x} \\ & \\ du=1 & v=-e^{-x} \end{array}\right]$ Integration by parts
$\displaystyle{V=2\pi \left([-e^{-x}]_{-1}^0\right)-2\pi[-xe^{-x}]_{-1}^0+2\pi\int_{-1}^0-e^{-x}\ dx}\\
\displaystyle{V=2\pi \left(-1+e\right)-2\pi \left(-1\right)}\\
\displaystyle{V=2\pi e}$
