Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/23) | Q#2

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  Question

The diagram shows the part of the curve y = xe-x. The shaded region R is bounded by the curve and by the lines x = 2, x = 3 and y = 0. 

     i.       Use the trapezium rule with two intervals to estimate the area of R, giving your answer correct  to 2 decimal places.

   ii.       State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate  of the true value of the area of R.

Solution

     i.
 

We are given that shaded region R is bounded by the curve and by the lines x = 2, x = 3 and y = 0. 

We are required to estimate area of shaded region R by applying Trapezium Rule.

Therefore, 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;

0

1

2

Therefore;

   ii.
 

If the graph is bending upwards over the whole interval from  to , then trapezium rule will give an  overestimate of the true area (as shown in the diagram below). 

It is evident that for the given graph trapezium rule will give an overestimate.

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