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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

0

1
dx 
1x2  12 2
˛

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
M
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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

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