Question

The diagram shows part of the curve  and three points A, B and C on the  curve with x-coordinates 1, 2 and 5 respectively.

i.       A point P moves along the curve in such a way that its x-coordinate increases at  a constant rate of 0.04 units per second. Find the rate at which the y-coordinate of P  is changing as P passes through A.

ii.       Find the volume obtained when the shaded region is rotated through  about
the x-axis.

Solution

i.
 

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

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

Since we are interested in rate of change of y-coordinate of P at point A, we,  therefore, first need the derivative of the curve at point A.

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;

For the given case;

Rule for differentiation of  is:

Rule for differentiation of  is:

Now, gradient of the curve at point A can be found by substituting x-coordinate of  point A given as    in derivative of the curve.

ii.
 

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

For the given case;

We have the algebraic formula;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is: