# 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 360o 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 cm3, of water is given by  . Water is poured into the bowl at a  constant rate of 2 cms-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 cm3 per second. This  translates to change in volume of water at a constant rate of 2 cm3 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 .