C2 June 2008 Q2

EdexcelOld spec6 marksIntegration

2. \[y = \sqrt{(5^x + 2)}\]

(a) Complete the table below, giving the values of \(y\) to 3 decimal places.

\(x\)	0	0.5	1	1.5	2
\(y\)			2.646	3.630

(2)

(b) Use the trapezium rule, with all the values of \(y\) from your table, to find an approximation for the value of \(\displaystyle\int_0^2 \sqrt{(5^x + 2)}\,\mathrm{d}x\). (4)

Mark scheme (a)

Scheme	Marks
1.732, 2.058, 5.196 awrt (One or two correct B1 B0, All correct B1 B1)	B1 B1
	(2)

Notes

Accept awrt (but less accuracy loses these marks).
Also accept exact answers, e.g. \(\sqrt{3}\) at \(x = 0\), \(\sqrt{27}\) or \(3\sqrt{3}\) at \(x = 2\).

Mark scheme (b)

Scheme	Marks
\(\dfrac{1}{2}\times 0.5\ \ldots\ldots\)	B1
\(\ldots\ldots\left\{(1.732 + 5.196) + 2(2.058 + 2.646 + 3.630)\right\}\)	M1 A1ft
\(= 5.899\) (awrt 5.9, allowed even after minor slips in values)	A1
	(4)
	6

Notes

For the M mark, the first bracket must contain the ‘first and last’ values, and the second bracket 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. \(\dfrac{1}{2}\times 0.5(1.732 + 5.196) + 2(2.058 + 2.646 + 3.630)\)
scores B1 M1 A0 A0 unless the final answer implies that the calculation has been done correctly (then full marks can be given).

\(x\) values: M0 if the values used in the brackets are \(x\) values instead of \(y\) values.

Alternative:
Separate trapezia may be used, and this can be marked equivalently.
\(\left[\dfrac{1}{4}(1.732 + 2.058) + \dfrac{1}{4}(2.058 + 2.646) + \dfrac{1}{4}(2.646 + 3.630) + \dfrac{1}{4}(3.630 + 5.196)\right]\)