Please, help with this integration problem Consider the region bounded by the curves $y=e^x$, $y=e^{-x}$, and $x=1$. Use the method of cylindrical shells to find the volume of the solid obtained by rotating this region about the y-axis.
I drew the corresponding graph. I'm confused by the fact that the area is rotating about one of the bounding lines (x=1) that bound the region (I have solved volumes this way, until now), but about the y-axis, which is a previous point to the bounding line. How does this changes the procedure?
Thank you!

Rotating about the vertical line $x=1$ is in principal no different than rotating about the $y$-axis, which after all is just the vertical line $x=0$. The radius of a shell when rotating about the $y$-axis is the distance of $x$ from $0$, which is $r=|x-0|=|x|$; this further simplifies to $|x|=x$ if $0\leq x$. The only modification required when rotating about a different vertical line such as $x=1$ is that now the radius of a shell is going to be the distance of $x$ from $1$, which is $r=|x-1|$, which will further simplify to $|x-1|=1-x$ if $x\leq 1$, as is the case for your problem.
