C4 June 2010 Q1

EdexcelOld spec8 marksIntegrationNumerical Methods

1.

Figure 1: curve decreasing gently from the y-axis; region R shaded between the curve, the axes and the line x = π/3
Figure 1

Figure 1 shows part of the curve with equation \(y = \sqrt{(0.75 + \cos^2 x)}\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis, the \(x\)-axis and the line with equation \(x = \dfrac{\pi}{3}\).

(a) Complete the table with values of \(y\) corresponding to \(x = \dfrac{\pi}{6}\) and \(x = \dfrac{\pi}{4}\).
\(x\)0\(\dfrac{\pi}{12}\)\(\dfrac{\pi}{6}\)\(\dfrac{\pi}{4}\)\(\dfrac{\pi}{3}\)
\(y\)1.32291.29731
(2)
(b) Use the trapezium rule
(i) with the values of \(y\) at \(x = 0\), \(x = \dfrac{\pi}{6}\) and \(x = \dfrac{\pi}{3}\) to find an estimate of the area of \(R\). Give your answer to 3 decimal places.
(ii) with the values of \(y\) at \(x = 0\), \(x = \dfrac{\pi}{12}\), \(x = \dfrac{\pi}{6}\), \(x = \dfrac{\pi}{4}\) and \(x = \dfrac{\pi}{3}\) to find a further estimate of the area of \(R\). Give your answer to 3 decimal places. (6)
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