\includegraphics{figure_4}

The diagram shows the part of the curve $y = 3x \sin 2x$ for which $0 \leqslant x \leqslant \frac{1}{2}\pi$.

The maximum point on the curve is denoted by $P$.

\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinate of $P$ satisfies the equation $\tan 2x + 2x = 0$. [3]

\item Use the Newton-Raphson method, with a suitable initial value, to find the $x$-coordinate of $P$, giving your answer correct to 4 decimal places. Show the result of each iteration. [4]

\item The trapezium rule, with four strips of equal width, is used to find an approximation to
$$\int_0^{\frac{1}{2}\pi} 3x \sin 2x \, dx.$$

Show that the result can be expressed as $k\pi^2(\sqrt{2} + 1)$, where $k$ is a rational number to be determined. [4]

\item \begin{enumerate}[label=(\roman*)]
\item Evaluate $\int_0^{\frac{1}{2}\pi} 3x \sin 2x \, dx$. [1]

\item Hence determine whether using the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region enclosed by the curve $y = 3x \sin 2x$ and the $x$-axis for $0 \leqslant x \leqslant \frac{1}{2}\pi$. [1]

\item Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region in this case. [1]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{OCR H240/03 2019 Q4 [14]}}