# 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:      