# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#10

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**Question**

The diagram shows the line and the curve , meeting at and .

** i. **Find the area of the shaded region.

** ii. **Find the volume obtained when the shaded region is rotated through 360◦ about the y-axis.

**Solution**

** i.
**

It is evident from the diagram that;

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

Rule for integration of is:

Rule for integration of is:

Rule for integration of is:

ii.

It is evident from the diagram that;

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

We have equations of curve and line as;

To find volume from rotation about y-axis we need to rewrite these equations in terms of y.

First equation of curve;

Now equation of line;

Now we can find the required volume;

Rule for integration of is:

Rule for integration of is:

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