# 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

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;