Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 58

Find the volume generated by rotating the region bounded by $y = e^x$,$ y =e^{-x}$ and $x = 1$ about $y$-axis. Use cylindrical shells method.
By using vertical strips have distance to the $y$-axis as $x$. Such that if you rotate these length about the $y$-axis, you'll get a circumference of $c = x$. Also, the height of the strips resembles the height of the cylinder as $H = y_{\text{upper}} - y_{\text{lower}} = e^x - e^{-x}$, Thus, the value is



$
\begin{equation}
\begin{aligned}
V &= \int^1_0 c (x) H(x) dx\\
\\
V &= \int^1_0 (2 \pi x) \left(e^x - e^{-x}\right) dx\\
\\
V &= 2 \pi \int^1_0 x \left(e^x - e^{-x}\right) dx
\end{aligned}
\end{equation}
$

By using integration by parts
$\displaystyle V = \frac{4\pi}{e}$ cubic units