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]
\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.
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]
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.
State whether \(A _ { 0 }\), your answer to part (i), is an underestimate for \(A\) or an overestimate. Give a reason for your answer.
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 \(A\) ?
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 \(A _ { 2 }\).