Volume Exponential Function

Question

I should find the Volume received by rotating the region bounded by:
$y = e^x $, $ y = 0 $,$ x = 0 $, $ x = 1 $ rotated around the x axis.
I know how to find it by using the disc method but I could not find a way to do it with the shell one.

Answer 1

When we rotate the part from $y=0$ to $y=1$ about the $x$-axis, we get a cylinder. The volume of the cylinder can be found using cylindrical shells, but can be found more simply.

As to the part between $y=1$ and $y=e$, the radius of the cylindrical shell at height $y$ is $y$. The "height" of the cylinder is $1-x$. In terms of $y$ it is $1-\ln y$. So the volume obtained by rotating that part is $$\int_{y=1}^e 2\pi y(1-\ln y)\,dy.$$ Use integration by parts, $u=1-\ln y$, $dv=y\,dy$. The disk method is more pleasant!