9 Fig. 9 shows
\(P \quad\) The line \(y = x\)
\(Q\) The curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\)
\(R \quad\) The curve \(\quad y = \sqrt { x }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c52d6b5-84b4-455a-9620-c377ae457069-4_471_1103_762_374}
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\caption{Fig. 9}
\end{figure}
- Write down the area of the triangle formed by the line \(y = x\), the line \(x = 1\) and the \(x\)-axis.
- Show that the area of the region formed by the curve \(y = \sqrt { x }\), the line \(x = 1\) and the \(x\)-axis is \(\frac { 2 } { 3 }\).
An estimate is required of the Area, \(A\), of the region formed by the curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\), the line \(x = 1\) and the \(x\)-axis.
- Use results to parts (i) and (ii) to complete the statement
$$\ldots \ldots \ldots \ldots . . < A < \ldots \ldots \ldots \ldots \ldots . .$$
- Use the Trapezium Rule with 4 strips to find an estimate for \(A\).
- Draw a sketch of Fig. 9. Use it to illustrate the area found as the trapezium rule estimate for \(A\).
Explain how your diagram shows that the trapezium rule estimate must be:
consistent with the answer to part (iv);
an under-estimate for A .