6
\includegraphics[max width=\textwidth, alt={}, center]{8a3f8707-67a4-4069-aba5-7e9496cb1748-10_588_583_260_781} The diagram shows the straight line \(x + y = 5\) intersecting the curve \(y = \frac { 4 } { x }\) at the points \(A ( 1,4 )\) and \(B ( 4,1 )\). Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.