Standard +0.8 This question requires finding intersection points of two curves involving parameter k, setting up a definite integral with correct limits, integrating two power functions, and then showing the parameter k cancels to give a constant answer. The algebraic manipulation and proof that the result is independent of k elevates this above a routine area calculation, but the techniques themselves are standard A-level integration.
\includegraphics{figure_7}
Figure 7 shows the curves with equations
$$y = kx^2 \quad x \geq 0$$
$$y = \sqrt{kx} \quad x \geq 0$$
where \(k\) is a positive constant.
The finite region \(R\), shown shaded in Figure 7, is bounded by the two curves.
Show that, for all values of \(k\), the area of \(R\) is \(\frac{1}{3}\) [5]
\includegraphics{figure_7}
Figure 7 shows the curves with equations
$$y = kx^2 \quad x \geq 0$$
$$y = \sqrt{kx} \quad x \geq 0$$
where $k$ is a positive constant.
The finite region $R$, shown shaded in Figure 7, is bounded by the two curves.
Show that, for all values of $k$, the area of $R$ is $\frac{1}{3}$ [5]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q17 [5]}}