# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#7

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Question

The diagram shows parts of the curves  and , intersecting at points A and  B.

i.       State the coordinates of A.

ii.       Find, showing all necessary working, the area of the shaded region.

Solution

i.

It is evident that point A is the intersection point of the two curves given by equations;

It is also evident from the equation that point A is the x-intercept of both the given curves.

Therefore, we can find the coordinates of point A as x-intercept of a curve rather easily.

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).

We choose;

Substituting ;

Hence, coordinates of .

ii.

It is evident from the diagram that;

To find the area of region under the curve , we need to integrate the curve from point  to   along x-axis.

We are given equation of curve 1 as;

We are given equation of curve 2 as;

Therefore;

It is evident from the diagram that for the shaded region x varies from  to .

Therefore;

Rule for integration of  is: