Question 10 - A-Level Maths

\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4782d612-0ec1-418e-8ef3-c871dce82b44-16_451_442_269_888} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Fig. 1 shows part of the curve \(y = x ^ { 2 } - 1\) and the line \(y = h\), where \(h\) is a constant.
1. The shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Show that the volume of revolution, \(V\), is given by \(V = \pi \left( \frac { 1 } { 2 } h ^ { 2 } + h \right)\).
2. Find, showing all necessary working, the area of the shaded region when \(h = 3\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4782d612-0ec1-418e-8ef3-c871dce82b44-17_257_408_1126_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Fig. 2 shows a cross-section of a bowl containing water. When the height of the water level is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water is given by \(V = \pi \left( \frac { 1 } { 2 } h ^ { 2 } + h \right)\). Water is poured into the bowl at a constant rate of \(2 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), at which the height of the water level is increasing when the height of the water level is 3 cm .