Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2018  OctNov  (P29709/23)  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 xaxis. 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 xaxis.
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 xaxis.
Expression for the volume of the solid formed when the shaded region under the curve is rotated completely about the xaxis 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:
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