Challenging +1.8 This is a solid of revolution problem requiring students to find where the function crosses the x-axis (x=1 and x=3), set up the integral V=π∫₁³[(x-3)√(ln x)]² dx, expand to (x-3)²ln x, and integrate by parts twice. The algebraic manipulation and repeated integration by parts with logarithmic functions makes this substantially harder than routine volume of revolution questions, though it follows a standard framework once set up correctly.
\includegraphics{figure_12}
The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the \(x\)-axis is rotated by \(2\pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\).
Determine the exact volume of \(S\). [7]
\includegraphics{figure_12}
The figure shows part of the graph of $y = (x - 3)\sqrt{\ln x}$. The portion of the graph below the $x$-axis is rotated by $2\pi$ radians around the $x$-axis to form a solid of revolution, $S$.
Determine the exact volume of $S$. [7]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q12 [7]}}