solid of revolution Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.
$y= 27e^x$, $y= 27e^{-x}$, $x = 1$.

Consider the approximately rectangular vertical strip running from the curve $y=27e^{-x}$ to the curve $y=27e^x$ at a distance $x$ from the $y$-axis and of width $\delta x.$ The height of the strip is about $27(e^x-e^{-x})$ and so its area is approximately
$$27(e^x-e^{-x}) \delta x.$$
Now consider rotating this about the $y$-axis one complete revolution. The rectangle will sweep out a volume of approximately
$$ 2 \pi x \times 27(e^x-e^{-x}) \delta x.$$
You then need to sum up all these elements of volume from $x=0$ to $x=1$ in the limit as $\delta x \rightarrow 0.$ This gives you the required integral.
Be sure to draw a diagram to see what is going on.
