Standard +0.3 This is a straightforward volume of revolution question about the y-axis using the standard formula V = π∫x²dy. The curve is already expressed as x in terms of y, requiring only substitution, expansion of the squared term, and integration of polynomial terms between given limits. It's slightly above average difficulty due to the algebraic manipulation needed when squaring (12/y² - 2), but remains a standard textbook exercise with no conceptual challenges.
3
\includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_497_1106_554_515}
The diagram shows part of the curve \(x = \frac { 12 } { y ^ { 2 } } - 2\). The shaded region is bounded by the curve, the \(y\)-axis and the lines \(y = 1\) and \(y = 2\). Showing all necessary working, find the volume, in terms of \(\pi\), when this shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.
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\includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_497_1106_554_515}
The diagram shows part of the curve $x = \frac { 12 } { y ^ { 2 } } - 2$. The shaded region is bounded by the curve, the $y$-axis and the lines $y = 1$ and $y = 2$. Showing all necessary working, find the volume, in terms of $\pi$, when this shaded region is rotated through $360 ^ { \circ }$ about the $y$-axis.
\hfill \mbox{\textit{CAIE P1 2016 Q3 [5]}}