# 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 = x^{2} + 4x + 3 for which x ≥ −2.

**
i. **Express y=x

^{2}+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 360^{O} 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:

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