Pie is Everywhere

Calculus Level 2

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.

It is then shot straight into a very thick wall (i.e. it does not pierce 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 \(i\).

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