SPS SPS SM Pure 2023 June — Question 17

Exam BoardSPS
ModuleSPS SM Pure (SPS SM Pure)
Year2023
SessionJune
TopicAreas Between Curves

17. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f953fd1c-447f-4e97-a42a-5264c053fda0-36_894_899_244_607} \captionsetup{labelformat=empty} \caption{Figure 7}
\end{figure} Figure 7 shows the curves with equations $$\begin{aligned} & y = k x ^ { 2 } \quad x \geqslant 0
& y = \sqrt { k x } \quad x \geqslant 0 \end{aligned}$$ 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 \(\frac { 1 } { 3 }\)