Pie is Everywhere

A bullet is formed by revolving the area bounded by the the curve $y = \ln(x)$from $x = 1$to $x = e$about the $x$-axis.

然后直接进入一个非常厚壁(i.e. it does not pierce through the other side at all) making a closed cylindrical hole until it stops moving. Then the bullet is carefully extracted without affecting the hole at all, leaving an empty hole with a pointy end where the bullet once was.

The length of the entire hole is $e+1$. If the volume of the hole can be expressed as $\pi i e,$where $i$is a constant, find the value of $i$.