Cambridge International AS and A Level Mathematics Pure Mathematics 1

Exercise

(^) 6H
P1^
6
6  The graph shows the curve y = x^2 − 4. The region R is formed by the line
y = 12, the x axis, the y axis and the curve y = x^2 − 4 for positive values of x.
(i) Copy the sketch graph and shade the region R.
The inside of a vase is formed by rotating the region R through 360° about the
y axis. Each unit of x and y represents 2 cm.
(ii) Write down an expression for the volume of revolution of the region R
about the y axis.
(iii) Find the capacity of the vase in litres.
(iv) Show that when the vase is filled to 56 of its internal height it is
three-quarters full.
[MEI]
7  The diagram shows the curve yx= 3
(^14)

. The shaded region is bounded by the
curve, the x axis and the lines x = 1 and x = 4.

Find the volume of the solid obtained when this shaded region is rotated

completely about the x axis, giving your answer in terms of π.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q2 June 2007]

–4 O 4

12

y

–2 2 x

–4

y

O^14 x

y = 3x