# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | May-Jun | (P1-9709/13) | Q#10

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**Question**

**a. ** Fig. 1 shows part of the curve and the line y = h, where h is a constant.

** (i) **The shaded region is rotated through 360^{o} about the y-axis. Show that the volume of revolution, V, is given by

** (ii) **Find, showing all necessary working, the area of the shaded region when h = 3.

**b. **

Fig. 2 shows a cross-section of a bowl containing water. When the height of the water level is h cm, the volume, V cm^{3}, of water is given by . Water is poured into the bowl at a constant rate of 2 cm^{3 }s^{-1}. Find the rate, in cms^{-1}, at which the height of the water level is increasing when the height of the water level is 3 cm.

**Solution**

**(i)
**

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

We are given equation of the curve as;

We can write it in terms of instead of to find ;

Therefore;

It is evident from the diagram that shaded region extends along y-axis from to .

Therefore;

Rule for integration of is:

**(ii)
**

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

It is evident from the diagram that area of shaded region is area under the curve along y-axis from to . Therefore, we write equation of curve in terms of instead of to find area of shaded region;

Therefore;

Rule for integration of is:

**(iii) **

We are required to find the rate, in cm per second, at which the height of water level is rising when the height of the water level is 3 cm.

Rate of change of with respect to is derivative of with respect to ;

Rate of change of with respect to at a particular point can be found by substituting x- coordinates of that point in the expression for rate of change;

Therefore, we are required to find;

We are given that water is steadily poured into the bowl at a constant rate of 2 cm^{3} per second. This translates to change in volume of water at a constant rate of 2 cm^{3} per second. Therefore;

We know that;

Therefore;

Since we are looking for , we need;

We have from (i) that;

Therefore we can find ;

Rule for differentiation of is:

Rule for differentiation of is:

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that particular point.

Gradient (slope) of the curve at a particular point can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

Therefore;

We are given that , therefore;

Therefore, at we have .

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