# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | 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 x-axis.

It is evident that to find area of the shaded region we need x-coordinates 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 x-coordinates 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 x-axis () right under points D & C respectively, therefore;

Now we can find AB & AD;

Hence;

Finally;