IBHM 446 472 .pdf
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16 Integration 3 – Applications
b Find the values of t for which v 1t 2 0, given that 0
t
6.
c i Write down a mathematical expression for the total distance travelled
by the particle in the first six seconds after passing through O.
ii Find this distance.
[IB Nov 01 P2 Q2]
˛
✗ 13 When air is released from an inflated balloon it is found that the rate of
M
C
7
4
1
M–
M+
CE
%
8
9
–
5
6
÷
2
3
+
0
ON
X
=
decrease of the volume of the balloon is proportional to the volume of the
dv
kv,
balloon. This can be represented by the differential equation
dt
where v is the volume, t is the time and k is the constant of proportionality.
a If the initial volume of the balloon is v0, find an expression, in terms of k,
for the volume of the balloon at time t.
v0
b Find an expression, in terms of k, for the time when the volume is .
2
[IB May 99 P1 Q19]
˛
˛
✗ 14 Show by means of the substitution x tan u that
M
C
7
4
1
M–
M+
CE
%
8
9
–
5
6
÷
2
3
+
0
ON
X
=
1
冮
0
1
dx
1x2 12 2
˛
冮
p
4
cos2 u du. Hence find the exact value of the volume
0
1
bounded by the lines x 0 and x 1
x 1
formed when the curve y
2
˛
is rotated fully about the x -axis.
✗ 15 Consider the curve y
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7
4
1
0
M–
M+
CE
%
8
9
–
5
6
÷
2
3
+
ON
X
˛
2
9a 14a x2.
˛
=
a Sketch the part of the curve that lies in the first quadrant.
b Find the exact value of the volume Vx when this part of the curve is
rotated through 360° about the x-axis.
˛
Vy
˛
c Show that
Vx
˛
p
where Vy is the volume generated when the curve is
q
˛
rotated fully about the y-axis and p and q are integers.
472