# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#9

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:              