Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2010  MayJun  (P19709/11)  Q#4
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Question
The diagram shows the curve and the line . Find the area of the shaded region.
Solution
i.
Consider the diagram below.
It is evident that;
Therefore, first 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 xaxis.
It is evident that to find area of the shaded region we need xcoordinates of points of intersection of the curve and the line.
If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the line and the
curve).
Equation of the line is;
Equation of the curve is;
Equating both equations;
Now we have two options;






Two values of x indicate that there are two intersection points.
Now we can find area under the curve.
Rule for integration of is:
Rule for integration of is:
Now we find area of rectangle ABCD.
Expression for the area of the rectangle is;
For the given case;
Now we need to find AB & AD.
We have the area Consider the diagram below.
Expression to find distance between two given points and is:
Therefore, we need coordinates of points A, B & D.
We already have xcoordinates of points of intersection of curve and the line i.e. C & D.
It is evident from the diagram that;
Since the point D lies on line ;
Therefore;
Similarly, point A & B lie on xaxis () right under points D & C respectively, therefore;
Now we can find AB & AD;








Hence;
Finally;
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