# 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

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.