A calculus problem by Liam Milliken

Calculus Level pending

Say that we have a function \(f(\chi) = Re(-ln(\chi)+\frac{|\chi|}{-ln(\chi)})\). The solid of revolution given by rotating \((\kappa \cdot f)(\chi)\) about the \(\chi\) axis between \(-5\) and \(-e^3 \approx \pi \cdot 439.862\). What is \(\kappa\)?

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