Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2017  FebMar  (P29709/22)  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 xaxis.
For the given case;
Therefore;
It is evident from the diagram that shaded region extends from to xintercept point of the curve.
Therefore, we need to find the xcoordinate of the xintecrept of the curve.
The point at which curve (or line) intercepts xaxis, 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 xintercept points of the curve. However, the shaded region extends from to first xintercept 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:
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