C4 January 2008 Q1

EdexcelOld spec6 marksIntegration

Figure 1: curve y = e^x root(sin x) for 0 to pi, with region R shaded between the curve and the x-axis — Figure 1

The curve shown in Figure 1 has equation \(y = \mathrm{e}^x\sqrt{(\sin x)}\), \(0 \leqslant x \leqslant \pi\). The finite region \(R\) bounded by the curve and the \(x\)-axis is shown shaded in Figure 1.

(a) Complete the table below with the values of \(y\) corresponding to \(x = \dfrac{\pi}{4}\) and \(\dfrac{\pi}{2}\), giving your answers to 5 decimal places.

\(x\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)
\(y\)	0			8.87207	0

(2)

(b) Use the trapezium rule, with all the values in the completed table, to obtain an estimate for the area of the region \(R\). Give your answer to 4 decimal places. (4)

x	0	\(\frac{\pi}{4}\)	\(\frac{\pi}{2}\)	\(\frac{3\pi}{4}\)	\(\pi\)
y	0	1.844321332…	4.810477381…	8.87207	0

Scheme	Marks
awrt 1.84432	B1
awrt 4.81048 or 4.81047	B1
	(2)

Mark scheme (b)

Way 1

Scheme	Marks
Area \(\approx\) \(\dfrac{1}{2} \times \dfrac{\pi}{4};\ \times \underline{\left\{0 + 2(1.84432 + 4.81048 + 8.87207) + 0\right\}}\)	B1 M1ft A1ft
\(= \dfrac{\pi}{8} \times 31.05374\ldots = 12.19477518\ldots = \underline{12.1948}\) (4dp)	A1 cao
	(4)
	(6 marks)

Notes

B1: Outside brackets awrt 0.39 or \(\tfrac{1}{2} \times\) awrt 0.79 \(\tfrac{1}{2} \times \tfrac{\pi}{4}\) or \(\tfrac{\pi}{8}\)

M1ft: For structure of trapezium rule \(\{\ldots\ldots\ldots\}\); (0 can be implied)

A1ft: Correct expression inside brackets which all must be multiplied by their “outside constant”.

A1 cao: 12.1948

Aliter (b) Way 2

Scheme	Marks
Area \(\approx \tfrac{\pi}{4} \times \left\{\tfrac{0+1.84432}{2} + \tfrac{1.84432+4.81048}{2} + \tfrac{4.81048+8.87207}{2} + \tfrac{8.87207+0}{2}\right\}\)	B1
which is equivalent to: Area \(\approx\) \(\dfrac{1}{2} \times \dfrac{\pi}{4};\ \times \underline{\left\{0 + 2(1.84432 + 4.81048 + 8.87207) + 0\right\}}\)	M1ft A1ft
\(= \dfrac{\pi}{4} \times 15.52687\ldots = 12.19477518\ldots = \underline{12.1948}\) (4dp)	A1 cao
	(4)

B1: \(\tfrac{\pi}{4}\) (or awrt 0.79) and a divisor of 2 on all terms inside brackets.

M1ft: One of first and last ordinates, two of the middle ordinates inside brackets ignoring the 2. A1ft: Correct expression inside brackets if \(\tfrac{1}{2}\) was to be factorised out. A1 cao: 12.1948

Note an expression like Area \(\approx \dfrac{1}{2} \times \dfrac{\pi}{4} + 2(1.84432 + 4.81048 + 8.87207)\) would score B1M1A0A0