6 The diagram shows the curve with equation $y = \sqrt { 100 - 4 x ^ { 2 } }$, where $x \geqslant 0$.\\
\includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_518_494_367_758}
\begin{enumerate}[label=(\alph*)]
\item Calculate the volume of the solid generated when the region bounded by the curve shown above and the coordinate axes is rotated through $360 ^ { \circ }$ about the $\boldsymbol { y }$-axis, giving your answer in terms of $\pi$.
\item Use the mid-ordinate rule with five strips of equal width to find an estimate for $\int _ { 0 } ^ { 5 } \sqrt { 100 - 4 x ^ { 2 } } \mathrm {~d} x$, giving your answer to three significant figures.
\item The point $P$ on the curve has coordinates $( 3,8 )$.
\begin{enumerate}[label=(\roman*)]
\item Find the gradient of the curve $y = \sqrt { 100 - 4 x ^ { 2 } }$ at the point $P$.
\item Hence show that the equation of the tangent to the curve at the point $P$ can be written as $2 y + 3 x = 25$.
\end{enumerate}\item The shaded regions on the diagram below are bounded by the curve, the tangent at $P$ and the coordinate axes.\\
\includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_642_546_1800_731}

Use your answers to part (b) and part (c)(ii) to find an approximate value for the total area of the shaded regions. Give your answer to three significant figures.
\end{enumerate}

\hfill \mbox{\textit{AQA C3 2009 Q6 [19]}}