Standard +0.8 This is a solid of revolution problem requiring integration of multiple sections (line segments and a circular arc) rotated about the y-axis. Students must identify the correct radius functions for each region, set up multiple integrals with appropriate limits, and handle the circular arc equation. While the techniques are standard Further Maths content, the multi-part setup and coordinate manipulation elevate it above routine questions.
O is the origin of a coordinate system whose units are cm.
The points A, B, C and D have coordinates (1, 0), (1, 4), (6, 9) and (0, 9) respectively.
The arc BC is part of the curve with equation \(x^2 + (y - 10)^2 = 37\).
The closed shape OABCD is formed, in turn, from the line segments OA and AB, the arc BC and the line segments CD and DO (see diagram).
A funnel can be modelled by rotating OABCD by \(2\pi\) radians about the y-axis.
\includegraphics{figure_9}
Find the volume of the funnel according to the model. [3]
O is the origin of a coordinate system whose units are cm.
The points A, B, C and D have coordinates (1, 0), (1, 4), (6, 9) and (0, 9) respectively.
The arc BC is part of the curve with equation $x^2 + (y - 10)^2 = 37$.
The closed shape OABCD is formed, in turn, from the line segments OA and AB, the arc BC and the line segments CD and DO (see diagram).
A funnel can be modelled by rotating OABCD by $2\pi$ radians about the y-axis.
\includegraphics{figure_9}
Find the volume of the funnel according to the model. [3]
\hfill \mbox{\textit{SPS SPS FM 2023 Q9 [3]}}