# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/22) | Q#6

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Question

A.         Find B.

i.
Use the trapezium rule with 2 intervals to estimate the value of giving your answer correct to 3 decimal places.

ii.
Using a sketch of the graph of for , explain whether the trapezium rule  gives an under-estimate or an over-estimate of the true value of the integral in part (i).

Solution

A.

We are required to find; We have formula;   We have the trigonometric identity; Therefore;  We know that; Therefore; Hence; Rule for integration of is:  Rule for integration of is: Rule for integration of is:   B.

i.

We are required to apply Trapezium Rule to evaluate; The trapezium rule with intervals states that;  We are given that there are two intervals, .

We are also given that and .

Hence;        1   2  3  Therefore;   ii.

We know that graph of is as shown below. However, we are only interested in for the interval . Therefore; If the graph is bending upwards over the whole interval from to , then trapezium rule will give an  overestimate of the true area. It is evident that for the given graph trapezium rule will give an overestimate.