Finding volume via disk method rotated about the x-axis

Question

Find the volume generated by revolving the region bounded by the graphs of $y=e^{-x}$, $y=0$, $x=0$, $y=1$ about the $x$-axis.

before attempting this problem I noticed the "bounded region" is not fully closed as all the graphs are able to approach positive infinity because $y=e^{-x}$ has the $x$-axis as an asymptote meaning it does not have any roots. However, my book's answer key claims that there is indeed a value for this volume and it is approximately $1.358$.

Answer 1

hint: but volume might still approch a limit. just set the integral under proper limits u shd get the answer

Answer 2

Note that the bounds correspond to $0<x<\infty.$ Then the volume you seek is \pi $\int_0^\infty e^{-2x}=\pi/2\approx 1.57$. So I would think your book is wrong.