# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2018 | Oct-Nov | (P2-9709/23) | Q#6

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Question

The diagram shows the curve  for . The region R is  bounded by the curve, the axes and the line .

i.       Use the trapezium rule with two intervals to find an approximation to the area of  R, giving your answer correct to 3 significant figures.

ii.       The region R is rotated completely about the x-axis. Without using a calculator,  find the exact volume of the solid produced.

Solution

i.

We are required to find an estimate of the area of shaded region bounded by the  curve for , the axes and the line .

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

Therefore;

The trapezium rule with  intervals states that;

We are given that there are two intervals, .

We are also given that and .

Hence;

 1 2 3

Therefore;

ii.

We are required to find the exact volume of the solid produced when region R is  rotated completely about the 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 the curve has equation  for  .

Therefore;

Therefore, it can be written as;

For the given case;

Hence;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is: