# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | Oct-Nov | (P2-9709/21) | Q#7

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Question The diagram shows the curve y = e– x. The shaded region R is bounded by the curve and the lines y = 1 and x = p, where p is a constant.

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

ii. Show that if the area of R is equal to 1 then p = 2 – e-p.

iii. Use the iterative formula with initial value p1 = 2, to calculate the value of p correct to 2 decimal places. Give the result of  each iteration to 4 decimal places.

Solution

i.

We are required to find area of shaded region R.

It is evident that area of shaded region R is given by; First we find area of rectangle.

Expression for the area of the rectangle is; Therefore; Next we find area under the curve.

To find the area of region under the curve , we need to integrate the curve from point to along x-axis. For the given case; Therefore; Rule for integration of , or ;      Finally, we can find area of shaded region R as;    ii.

We are given that area of shaded region R is 1.

We have demonstrated in (i) that; Therefore;   iii.

If the sequence given by the inductive definition , with some initial value , converges  to a limit , then is the root of the equation .

Therefore, if , then is a root of .

Therefore, iterative formula we use; We use as initial value.   1  2  3  4  5  6  7  8  It is evident that .

Hence, is a root of .

The root given correct to 2 decimal places is 1.84.