October 2021 Paper 2 Q6

OCR ACurrent spec5 marksIntegration

6 Alex is investigating the area, \(A\), under the graph of \(y = x^2\) between \(x = 1\) and \(x = 1.5\). They draw the graph, together with rectangles of width \(\delta x = 0.1\), and varying heights \(y\).

Graph of y = x squared from x = 1 to x = 1.5 with five strips of width 0.1 marked at 1, 1.1, 1.2, 1.3, 1.4, 1.5; each strip has a lower rectangle under the curve and an upper rectangle above it, drawn with dashed lines

(a) Use the rectangles in the diagram to show that lower and upper bounds for the area \(A\) are 0.73 and 0.855 respectively. [1]

(b) Alex finds lower and upper bounds for the area \(A\), using widths \(\delta x\) of decreasing size.
The results are shown in the table. Where relevant, values are given correct to 3 significant figures.

Width \(\delta x\)	0.1	0.05	0.025	0.0125
Lower bound for area \(A\)	0.73	0.761	0.776	0.784
Upper bound for area \(A\)	0.855	0.823	0.807	0.799

Use Alex’s results to estimate the value of \(A\) correct to 2 significant figures. Give a brief justification for your estimate. [2]

Scheme	Marks
\(0.1(1 + 1.1^2 + 1.2^2 + 1.3^2 + 1.4^2)\) \(0.1(1.1^2 + 1.2^2 + 1.3^2 + 1.4^2 + 1.5^2)\)	B1
	[1]

Mark scheme (b)

Scheme	Marks
0.79	B1
About half way between the last two bounds or \((0.784 + 0.799) \div 2 = 0.79\) Ignore all else	B1
	[2]

Notes

B1: Not 0.7915

B1: condone "The mean of the last two bounds" or other sensible
Allow UB and LB are converging towards 0.79 oe

The two B1 marks are independent

Mark scheme (c)

Scheme	Marks
\(\displaystyle\lim_{\delta x \to 0} \sum_{x=1}^{1.5} y\,\delta x\)	B1 B1
	[2]

Notes

B1 for \(\displaystyle\lim_{\delta x \to 0} \sum y\,\delta x\). Allow \(x^2\) instead of \(y\)

B1 for limits, dep using \(\Sigma\) not integral. \(\displaystyle\lim_{\delta x \to 0} \sum_{1}^{1.5} y\,\delta x\) B1B0