Find the volume of the solid

Question

Find the volume of the solid by revolving the region

Answer 1

For this problem it helps to take a look at the graph of y=ln(x)/x. You will see the graph crosses the x-axis at x=1, goes a little bit up and then as x goes to infinity it approaches 0 (the axis).

The volume is computed using an improper integral:

V= π ∫₀^∞ (ln(x)/x)² dx.

To compute this integral, you first need to compute the indefinite integral ∫ (ln(x)/x)² dx. You rewrite (ln(x)/x)² as (ln(x))²/x² and then use integration by parts with u=(ln(x))² and dv=1/x².

I hope this helps, let me know if you want me to provide more details.