C2 June 2009 Q4

EdexcelOld spec8 marksIntegration

(a) Complete the table below, giving values of \(\sqrt{(2^x + 1)}\) to 3 decimal places.

\(x\)	0	0.5	1	1.5	2	2.5	3
\(\sqrt{(2^x + 1)}\)	1.414	1.554	1.732	1.957			3

(2)

Figure 1: the curve y = √(2^x + 1) with the region R under it shaded between x = 0 and x = 3 — Figure 1

Figure 1 shows the region \(R\) which is bounded by the curve with equation \(y = \sqrt{(2^x + 1)}\), the \(x\)-axis and the lines \(x = 0\) and \(x = 3\)

(b) Use the trapezium rule, with all the values from your table, to find an approximation for the area of \(R\). (4)

(c) By reference to the curve in Figure 1 state, giving a reason, whether your approximation in part (b) is an overestimate or an underestimate for the area of \(R\). (2)

Scheme	Marks
\(x = 2\) gives 2.236 (allow AWRT) Accept \(\sqrt{5}\)	B1
\(x = 2.5\) gives 2.580 (allow AWRT) Accept 2.58	B1
	(2)

Mark scheme (b)

Scheme	Marks
\(\left(\dfrac{1}{2} \times \dfrac{1}{2}\right),\quad \left[(1.414 + 3) + 2(1.554 + 1.732 + 1.957 + 2.236 + 2.580)\right]\)	B1,[M1A1ft]
\(= 6.133\) (AWRT 6.13, even following minor slips)	A1
	(4)

Notes

B1 for \(\dfrac{1}{2} \times \dfrac{1}{2}\) or equivalent.

For the M mark, the first bracket must contain the 'first and last' values, and the second bracket (which must be multiplied by 2) must have no additional values. If the only mistake is to omit one of the values from the second bracket, this can be considered as a slip and the M mark can be allowed.

Bracketing mistake: i.e. \(\left(\dfrac{1}{2} \times \dfrac{1}{2}\right)(1.414 + 3) + 2(1.554 + 1.732 + 1.957 + 2.236 + 2.580)\)
scores B1 M1 A0 A0 unless the final answer implies that the calculation has been done correctly (then full marks can be given).

Alternative:
Separate trapezia may be used, and this can be marked equivalently.\[\left[\frac{1}{4}(1.414 + 1.554) + \frac{1}{4}(1.554 + 1.732) + \ldots\ldots\ldots\ldots + \frac{1}{4}(2.580 + 3)\right]\]1^st A1ft for correct expression, ft their 2.236 and their 2.580

Mark scheme (c)

Scheme	Marks
Overestimate	B1
'Since the trapezia lie above the curve', or an equivalent explanation, or sketch of (one or more) trapezia above the curve on a diagram (or on the given diagram, in which case there should be reference to this). (Note that there must be some reference to a trapezium or trapezia in the explanation or diagram).	dB1
	(2)
	[8]

Notes

1^st B1 for 'overestimate', ignoring earlier mistakes and ignoring any reasons given.
2^nd B1 is dependent upon the 1^st B1 (overestimate).