# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | May-Jun | (P2-9709/02) | Q#3

Question

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

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

ii.       Hence calculate the value of p for which the area of R is equal to 5. Give your answer correct  to 2 significant figures.

Solution

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=0 to x=p.

Therefore;

Rule for integration of , or ;

Therefore;

ii.

We are given that;

We have demonstrated in (i) that;

Therefore;

Taking natural log of both sides;

For  ,  , .

Therefore;