C2 June 2014 (R) Q3

EdexcelOld spec5 marksIntegration

Figure 1: curve y = √(2x − 1) with region R shaded between x = 2 and x = 10 — Figure 1

Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt{(2x - 1)}\), \(x \geqslant 0.5\)

The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 10\).

The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt{(2x - 1)}\).

\(x\)	2	4	6	8	10
\(y\)	\(\sqrt{3}\)		\(\sqrt{11}\)		\(\sqrt{19}\)

(a) Complete the table with the values of \(y\) corresponding to \(x = 4\) and \(x = 8\). (1)

(b) Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 2 decimal places. (3)

Scheme	Marks
\(\sqrt{7}\) and \(\sqrt{15}\)	B1
	(1)

Mark scheme (b)

Scheme	Marks
Area\((R) \approx \dfrac{1}{2} \times 2; \times \underline{\left\{\sqrt{3} + 2\left(\sqrt{7} + \sqrt{11} + \sqrt{15}\right) + \sqrt{19}\right\}}\)	B1; M1
Note decimal values are \(\dfrac{1}{2} \times 2; \times \underline{\left\{\sqrt{3} + \sqrt{19} + 2\left(\sqrt{7} + \sqrt{11} + \sqrt{15}\right)\right\}} = \dfrac{1}{2} \times 2; \times \underline{\left\{6.0909.. + 19.6707\ldots\right\}}\)
\(= 1 \times 25.76166865\ldots = 25.76166\ldots = \underline{25.76}\) (2dp)	A1 cao
	(3)

Notes

B1: Outside brackets \(\tfrac{1}{2} \times 2\) or 1 (may be implied)

M1: For structure of \(\{\ldots\ldots\ldots\ldots\}\)

M1 requires the correct structure for the \(y\) values. It needs to contain first \(y\) value plus last \(y\) value and the second bracket to be multiplied by 2 and to be the summation of the remaining \(y\) values in the table with no additional values. If the only mistake is a copying error or is to omit one value from 2(…..) bracket this may be regarded as a slip and the M mark can be allowed (nb: an extra repeated term, however, forfeits the M mark). M0 if any values used are \(x\) values instead of \(y\) values.

Bracketing mistakes: e.g.
\(\left(\dfrac{1}{2} \times 2\right) \times \left(\sqrt{3} + \sqrt{19}\right) + 2\left(\sqrt{7} + \sqrt{11} + \sqrt{15}\right)\)
\(\left(\dfrac{1}{2} \times 2\right) \times \sqrt{3} + \sqrt{19} + 2\left(\sqrt{7} + \sqrt{11} + \sqrt{15}\right)\)
Both score B1 M1

Alternative

Separate trapezia may be used, and this can be marked equivalently.
\(\left[\dfrac{1}{2} \times 2(\sqrt{3} + \sqrt{7}) + \dfrac{1}{2} \times 2(\sqrt{7} + \sqrt{11}) + \dfrac{1}{2} \times 2(\sqrt{11} + \sqrt{15}) + \dfrac{1}{2} \times 2(\sqrt{15} + \sqrt{19})\right]\)
B1 for \(\dfrac{1}{2} \times 2\), M1 for correct structure

A1 cao: 25.76

Scheme	Marks
underestimate	B1
	(1)
	Total 5