8
\begin{enumerate}[label=(\roman*)]
\item Evaluate $A _ { 0 } = \int _ { 0 } ^ { 2 } \left( 2 + 2 x - x ^ { 2 } \right) \mathrm { d } x$.

Fig 8.1 illustrates the cross-section of a proposed tunnel. Lengths are in metres. The equation of the curved section is $y = 2 + \sqrt { 2 x - x ^ { 2 } }$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{23771896-942c-4a1d-ab95-6b6d3cc5643c-3_419_515_1155_836}
\captionsetup{labelformat=empty}
\caption{Fig. 8.1}
\end{center}
\end{figure}

The designers need to know the area of the cross-section, $A \mathrm {~m} ^ { 2 }$, so that they can work out the volume of the soil that will need to be removed when the tunnel is built.
\item An initial estimate, $A _ { 1 }$, is given by the area of the 8 rectangles shown in Fig 8.2. Calculate $A _ { 1 }$, and state whether it is an overestimate or underestimate for $A$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{23771896-942c-4a1d-ab95-6b6d3cc5643c-3_520_645_2053_644}
\captionsetup{labelformat=empty}
\caption{Fig. 8.2}
\end{center}
\end{figure}
\item On graph paper, draw the graphs of

$$y = 2 + 2 x - x ^ { 2 } \text { and } y = 2 + \sqrt { 2 x - x ^ { 2 } } \text { for } 0 \leq x \leq 2 .$$

Make it clear which equation applies to which curve.
\item State whether $A _ { 0 }$, your answer to part (i), is an underestimate for $A$ or an overestimate. Give a reason for your answer.
\item The designers use the trapezium rule to estimate $A$. What values does this give when they take\\
(A) 2 strips,\\
(B) 4 strips,\\
(C) 8 strips?

What can you conclude about the value of $A$ ?
\item The best estimate from the trapezium rule is denoted by $A _ { 2 }$.

State, with a reason, whether the true value of $A$ is nearer $A _ { 1 }$ or $A _ { 2 }$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4  Q8 [18]}}