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.

It is evident that .

Hence, is a root of .

The root given correct to 2 decimal places is 1.84.