# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/23) | Q#6

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Question

a)   Find b)  Without using a calculator, find the exact value of giving your answer in the form .

c) The diagram shows the curve for 0 x 6. The region bounded by the curve and  the lines x = 0, x = 6 and y = 0 is denoted by R. Use the trapezium rule with 2 strips to find an  estimate of the area of R, giving your answer correct to 1 decimal place.

Solution

a)

We are required to find;     Rule for integration of is:  We integrate both parts and one by one.

First we integrate . Rule for integration of , or ;   Next we integrate; Rule for integration of , or ;   Hence; b)

We are required to find the exact value of Rule for integration of is:  Rule for integration of is: This integral is valid only when .     Division Rule;   Power Rule;    c)

We are required to find an estimate of the area of R bounded by the curve and the lines x = 0, x = 6 and y = 0. To find the area of region under the curve , we need to integrate the curve from point to along x-axis. Therefore; 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;    