Pie is Everywhere

Calculus Level 1

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

It is then shot straight into a very thick wall (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+1e+1. If the volume of the hole can be expressed as πie,\pi i e, where ii is a constant, find the value of ii.

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