# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Feb-Mar | (P2-9709/22) | Q#7

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Question The diagram shows part of the curve The shaded region is bounded by the curve and the two axes.

i. Show that can be expressed in the form where the values of the constants and are to be determined.

ii.Find the exact area of the shaded region.

Solution

i.

We are given;        Therefore;  Hence;   Therefore;  Hence;  ii.

We are required to find the exact area of the shaded region.

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; It is evident from the diagram that shaded region extends from to x-intercept point of the  curve.

Therefore, we need to find the x-coordinate of the x-intecrept of the curve.

The point at which curve (or line) intercepts x-axis, the value of . So we can find the  value of coordinate by substituting in the equation of the curve (or line).

Hence;  Now we have two options.           Two values of x indicate that there are two intersection points.

It is evident from the diagram that there are two x-intercept points of the curve. However, the shaded region extends from to first x-intercept point where .

Therefore; As demonstrated in (i) above given equation can be written as; Therefore;  Rule for integration of is:  Rule for integration of is: Rule for integration of is: Rule for integration of is:          