# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2002 | Oct-Nov | (P2-9709/02) | Q#6

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Question

a)   Find the value of

b)

The diagram shows part of the curve  . The shaded region R is bounded by the curve and by  the lines x =1, y = 0 and x = p.

i.
Find, in terms of p, the area of R.

ii.       Hence find, correct to 1 decimal place, the value of p for which the area of R is equal to 2.

Solution

a)

We are required to evaluate;

Rule for integration of  is:

Therefore;

Rule for integration of  is:

Rule for integration of  is:

b)

i.

To find the area of region under the curve , we need to integrate the curve from point  to   along x-axis.

It is evident from the diagram that area R of shaded region is area under the curve  along x- axis from x=1 to x=p.

Therefore;

Rule for integration of  is:

This integral is valid only when .

It is evident that for the given case  and , therefore, . Since the limits of the  integral is from x=1 to x=p, therefore, . Hence, integral is valid.

ii.

We are given that;

We have demonstrated in (b:i) that;

Therefore;

Division Rule;

Hence, we can write;

Taking inverse of both sides;

and  are inverse functions. The composite function  is an identity function, with  domain the positive real numbers. Therefore;

Therefore;