# 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

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** 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;

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