# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#11

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Question

The diagram shows a shaded region bounded by the y-axis, the line y = 1 and the part of the  curve  y = x2 + 4x + 3 for which x 2.

i.
Express y=x2+4x+3 in the form y=(x+a)2+b, where a and b are constants. Hence, for x 2,  express x in terms of y.

ii.       Hence, showing all necessary working, find the volume obtained when the shaded region is rotated through 360O about the y-axis.

Solution

i.

We have the expression;

We use method of “completing square” to obtain the desired form. We complete the square for the  terms which involve .

We have the algebraic formula;

For the given case we can compare the given terms with the formula as below;

Therefore, we can deduce that;

Hence, we can write;

To complete the square, we can add and subtract the deduced value of ;

Therefore, we can write;

Now we express in terms of .

ii.

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

Therefore, for the given case;

We have found in (i) that;

Hence, for to ;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is: