Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#9

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Question

The diagram shows part of the curve  which has a minimum point at M. The line  intersects the curve at the points A and B.

    i.        Find the coordinates of A, B and M.

   ii.       Find the volume obtained when the shaded region is rotated through 360 about the x-axis.

Solution


i.
 

It is evident from the diagram that points A & B are the points of intersection of the given line and the curve.

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.

Corresponding values of y coordinate can be found by substituting values of x in any of the two equation i.e either equation of the line or equation of the curve.

However, we notice that both points A & B lie on the line , therefore, their y-coordinates are same i.e. 5. It is evident from the diagram that;

Hence;

Now, to find the coordinates of point M we utilize the fact that M is the minimum point on the curve i.e. it is stationary point on the curve.

A stationary point  on the curve  is the point where gradient of the curve is equal to zero;

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is:

For the given case;

Therefore;

Rule for differentiation of  is:

Therefore;

Rule for differentiation of  is:

Hence;

Since point M is stationary point.

It is evident from the diagram that;

Corresponding values of y coordinate can be found by substituting values of x in equation of the curve.

Hence coordinates of point .


ii.
 

Consider the diagram below.

It is evident that volume under the line  from point A to point B when rotated around x-axis will form a cylinder with radius AC and height AB.

Expression to find distance between two given points  and is:

We have from (i),

We can also see that point C is right below point A therefore . Therefore,

Expression for the volume of the cylinder is;

Therefore;

It is evident from the diagram that;

We find out the volume formed by the rotation of area under curve around x-axis.

Expression for the volume of the solid formed when the shaded region under the curve  is rotated completely about the x-axis is;

We are given that;

Therefore;

Rule for integration of  is:



Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

Finally;

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