Standard +0.3 This is a straightforward volumes of revolution question requiring rearrangement of the hyperbola equation to express x² in terms of y, then applying the standard formula V = π∫x²dy. The algebra is routine (expanding and simplifying), and the integration involves only polynomial terms. Slightly above average difficulty due to the hyperbola context and algebraic manipulation, but still a standard C3 exercise.
9 The shape of a vase can be modelled by rotating the curve with equation \(16 x ^ { 2 } - ( y - 8 ) ^ { 2 } = 32\) between \(y = 0\) and \(y = 16\) completely about the \(\boldsymbol { y }\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_890_1210_1555_424}
The vase has a base.
Find the volume of water needed to fill the vase, giving your answer as an exact value.
9 The shape of a vase can be modelled by rotating the curve with equation $16 x ^ { 2 } - ( y - 8 ) ^ { 2 } = 32$ between $y = 0$ and $y = 16$ completely about the $\boldsymbol { y }$-axis.\\
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_890_1210_1555_424}
The vase has a base.\\
Find the volume of water needed to fill the vase, giving your answer as an exact value.
\hfill \mbox{\textit{AQA C3 2013 Q9 [5]}}