Answer
$\frac{\pi}{2}(1-\frac{1}{e^2})$
Work Step by Step
Setup the integration using disk method about the x-axis
$\pi \int_0^1 (e^{-x})^2dx$
$\pi \int_0^1 (e^{-2x})dx$, integrate
$-\frac{\pi}{2}[e^{-2x}]_0^1$, evaluate the definite integral
$-\frac{\pi}{2}(e^{-2} -1)$
$\frac{\pi}{2}(1-\frac{1}{e^2})$, simplify
