Standard +0.3 This is a straightforward integration problem requiring finding the maximum turning point via differentiation, then integrating a polynomial. The algebra is routine (expanding a cubic, differentiating, solving for critical point, integrating) with no conceptual challenges beyond standard A-level calculus techniques. The 6 marks reflect multiple steps rather than difficulty.
\includegraphics{figure_5}
Figure 5 shows a sketch of the curve \(C\) with equation \(y = (x - 2)^2(x + 3)\)
The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis.
Find the exact area of \(R\).
*(Solutions based entirely on graphical or numerical methods are not acceptable.)* [6]
\includegraphics{figure_5}
Figure 5 shows a sketch of the curve $C$ with equation $y = (x - 2)^2(x + 3)$
The region $R$, shown shaded in Figure 5, is bounded by $C$, the vertical line passing through the maximum turning point of $C$ and the $x$-axis.
Find the exact area of $R$.
*(Solutions based entirely on graphical or numerical methods are not acceptable.)* [6]
\hfill \mbox{\textit{SPS SPS SM 2020 Q13 [6]}}