Achievement Standard 90636
Integration 3.2
Achievement Questions
1. Find the indefinite integrals
a)  12x 4 dx
 6x
e)  8dx
c)
2
 4 x  2  dx
2
6
g)
1
 x 4 dx
i)
6
k)

3
  u  e  du
d)  x  x  5  dx
f)  x dx
b)
 x3 6 
h)    3  dx
 4 x 
1 

j)   x 
 dx
x

 x x2  
l)      dx
5 6 2
3
n)  5 dx
4x
x dx
x5 dx
m)  4x 1.6 dx
o)
  x  5
6
p)
dx
2. Integrate the rational expressions
 x4  2 x2 
a)  
dx
x


  7t  3
4
dt
 x6  x 4 
c)  
 dx
3
 x 
 5x3  x 
b)  
 dx
 x 
 5x2  4 x 
d)  
 dx
4
 x

 4 x5  3x3 
e)  
 dx
2
2
x


 x 4
f)  
dx
x


3. Integrate the exponential functions.
a)  e5 x dx
b)  6e 4 x dx
 5
d)   2 x
e

 dx

c)
 2e
e)
 e 
dx
f)  e 2 x 1  e x  dx
 4
g)    x
e

 dx

 e x  e3 x 
h)  
dx
x
 e

4 8 x
dx
4x 3
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4. Integrate
a)
5
b)    du
u
 1 
d)  
 dx
 4x  3 
 1 x 
f)   2  dx
 x 
1
 x dx
 1 
c)    dx
 5x 
 2 
e)  
 dx
 1 x 
5. Integrate the trigonometric expressions
a)  sin 5x dx
b)  3cos 2x dx
x
c)  5sin   dx
3
2
e)  sec  8x  dx
d)   cos  5  6x  dx
f)  cos ec2 x cot 2 x dx
3

h)  cos  x 
2

 4 x 
g)  sin 
 dx
 3 
6. Evaluate the following definite integrals.
  x  x  dx
d)  e dx
 8x  3 dx
c)   sin 4x  dx
a)
4
1
0.5

5
0
3
b)
2
0
2
5x
2
0
e)

 dx

f)
3x dx
5
4
3
2
1
30
10
20
1– 1
2
3
4
5
 12 
2  x 
 

6
7. Find the area between the x-axis, the curve y  x 2  6 and the lines x  0, x  4 as
shown.
y
30
20
10
– 1
Integration 2009
2
1
2
3
4
5 x
90636
5– 5
1
2
3
4
1
2
3
4
1– 1
2
3
4
8. Find the area between the x-axis and the curve y   x3  6 x 2  9 x from x  1 to x  3
5
4
3
2
1
1
2
3
4
5
– 1–
–
–
–
–
5
4
3
2
1
2
1
1– 2
2
3
4
5
1
y
1
2
4 x
3
9. Find the area between the x-axis and the curve y  e0.4 x as shown.
2
y
1
1
1–
– 1
1
3
2

22
– 2 – 1
1
2
3
4
5 x
10. Find the shaded area in the graph of y  sin x
1
y

1– 3
2
1
2
– 1

2
3
2
2 x
11. Find the total shaded area in the graph of y  x3  x 2  2 x
y
– 3
– 2
– 1
1
2
x
12. Find the volume generated when the line y  0.5 x is rotated about the x-axis between
x  0 and x  8 .
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y
4
2
1
5
4
3
2
4
8
12
1– 1
2
3
4
5
– 2
– 4
2
4
6
8
x
13. The function shown is y  12 x  3x 2 . Find the volume of the shape generated by
rotating the shaded area about the x-axis.
12
8
4
y
– 1
14. A function has derivative
when x  3
1
2
3
4
5
x
dy
 12 x  8 . Find an expression for y if y  20
dx
dy
 6 x 2  4 x . Find the equation of the curve if it passes
dx
through the point (3, 12).
15. A curve has derivative
16. The rate at which the volume of air contained in a heated balloon is changing is given
by
dV
 3t 2  4t  12
(V in cm3, t in seconds)
dt
If the volume at the start is 100cm3, find an expression for the volume at any time.
t (20  t )
.
20
Initially, the runner is at the starting point. Find an equation for the distance of the
runner from the starting point at any time. (time is in seconds, distance in metres)
How far is the runner from the staring point when she/he stops?
17. The velocity of a runner is given by v 
18. A car is accelerating at a rate given by a  12t .
When timing starts, the car is at point A and is travelling at 30m/sec.
Find an expression for the distance of the car from A at any time.
ds
 40 cos  60t  50  ,
19. The velocity of a point marked on a jigsaw blade is given by
dt
where s is the distance from the central point in centimetres and t is in seconds.
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The marked point on the blade is 2cm from the central point after 0.01 seconds.
Find an equation for the distance of the marked point at any time.
20. Given that
d2y
 24 x  6 , find a general solution for this differential equation.
dx 2
21. Find a general solution for each differential equation below.
dy
dy
 6x
 x2  2x
a)
b)
dx
dx
dy
dy 4
x
c) y
d)

dx
dx y
22. Find a general solution for the differential equation
dy x  6

dx y  2
23. Find a general solution for the differential equation
dy 6 y

dx
x
Write the answer in the form y = . . . .
24. Find a general solution for the differential equation 5
25. Find a general solution for the differential equation
dy
 6e 2 x
dx
dy
 6 xy
dx
26. Find the particular solution for the differential equation
dy
x
if the curve passes

dx 4 y
through the point (2, 8).
27. Find the particular solution for the differential equation 4
dy
 3e x if y = 12 when
dx
x=2
28. The cross-sectional area of a river is to be estimated using the trapezium rule. The
depth of the river at points 5m apart was measured from a bridge. The results are
given in the table. (x is the distance from the left bank of the river, d is the depth in
metres at that point.)
x
0
5
10
15
20
25
30
35
40
d
0
1.8 2.6 4.2 5.0 4.5 3.4 2.1
0
Use the trapezium rule to estimate the cross-sectional area of the river at this point.
29. Use the trapezium rule to estimate the area under the curve y  e x between x = 0 and
x = 3. Use an interval length of 0.5.
2
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30. The area under the curve y  xe x is to be estimated from x = 0 to x = 1.2.
Use the Trapezium rule with h = 0.2 to estimate this area.
31. An archway is 1.6m wide. The height of the archway at intervals of 0.2m is given in
the table.
x
0
0.2
0.4
0.6
0.80
1
1.2
1.4
1.6
height
1.56 1.82 1.98 2.04 1.88 1.66 1.46 1.34 1.26
Use Simpson’s Rule to estimate the area under the archway.
40
35
30
25
20
15
10
6
5
4
3
2
1
40
5
10
15
20
25
30
35
1
2
3
4
5
6
32. Use Simpson’s Rule to estimate the value of the definite integral

0.7
0.1
sin x dx
(Use h = 0.1)
33. The graph shows the temperature of a cooling metal plate over several hours. Use
Simpson’s Rule to estimate the area under the curve from t = 1 to t = 5 hours.
y
40
35
30
25
20
15
10
5
6
4
2
40
10
20
30
2
4
6
1
2
3
4
5
6
x
34. Part of a funnel is created by rotating the curve y   x  6  about the x-axis from
2
x  0 to x  6 . Find the volume of this shape.
40
y
30
20
10
2
35. Solve the differential equation
Integration 2009
4
6
x
3
dy cos 4 x
, given that y  0.5 when x 

8
dx
2y
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90636
Merit Questions
1. Integrate the following
a)  2sin  6 x  cos  2 x  dx
b)  2 cos  7 x  cos  3 x  dx
d)  6 cos  x    sin  x    dx
c)  sin  3 x  sin  x  dx
2. Integrate the following
a)  3x 2  x3  5  dx
b)  6 xe3 x dx
4
c)
2
d)  24 x  3x 2  8 dx
 2 x sin  x  dx
4
2
3. Use a suitable trigonometric identity to find the integral
4. Integrate the following
 3x 2 
a)   3
 dx
x

5


 tan x dx
6. Integrate the following
 4x  2 
a)  
 dx
 x3 
 6x 1 
b)  
 dx
 2 x 
7. Use the substitution given to integrate the following

1
1  x2
7
 4x
1  2 x 2 dx with u  1  2 x 2
dx
9. Use the substitution given to integrate the following
Integration 2009
x dx
 ex 
d)   x
 dx
 e  12 
sin x
, find the integral
cos x
8. By using the substitution x  sin u , find
2
 6x  4 
b)   2
 dx
 3x  4 x  1 
 12 x 
c)   2
 dx
 2x  3 
5. By writing tan x 
 sin
 4x 
 dx, with u  x  1
x 1 
 
90636
1
4
3
2
8
1– 3
2
3
4
1
2
 2 x2 
10. Use a suitable substitution to find  
 dx
 x 1 
8
y
11. Find the area enclosed between the
graphs of y  x 2  2 and y  x  4 .
– 3 – 2 – 1
1
2
3
12. Find the area enclosed between the graphs of y  x 2 and y  2 x  x 2
13. Find the area enclosed between the y-axis and the curve y  x 2  4 for y  4 to y  6
14. The velocity-time graph for a particle is shown. The functions are:
y  x3  1 0  x  2
y   x  5 2  x  5
Find the total distance travelled by the particle. This is given by the area under the
graph.
2
y
8
4
1
2
3
4
5
x
15. The curve y  1  x is rotated about the y-axis.
Find the volume generated if the curve is rotated between y  0 and y  1 .
16. The graph of y  log e x is rotated around the y-axis from y  0 to y  3 . Find the
volume of the shape formed.
12
is rotated around the x-axis from x  1 to x  10 . Find the
x
volume of the shape created.
17. The graph of y 
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4 x
18. Find the value of k where x  k is a vertical line which divides the area between
y  e2 x , x  0 , x  2 and the x-axis into two equal parts.
19. Show that y  x3  x is a solution to the differential equation
d2y
dy
x
 x  y  8 x3
2
dx
dx
2
dy
 y 2 e  x if y  1 when x  0 .
dx
Write your answer in the form y  ....
20. Solve the differential equation
21. The value of a machine changes at a rate which depends on its age. When the machine
is t years old, the rate at which its value V is changing is 250  t  7  dollars per year.
a) Find an expression for the value of the machine at any age if its initial value is
$6625.
b) What will it be worth when it is 7 years old.
22. The number of insects in a population after t years satisfies the equation
where k  0.05 and b  50 000 are constants.
The initial population is 1.1 million.
Find the equation for y and the population after 10 years.
dy
 ky  b
dt
23. The rate at which a radioactive substance decays is proportional to the amount of
radioactive material present.
a) If N is the amount of radioactive material at any time, write a differential equation
involving N and t.
b) If the initial amount of material is N0, write down the solution for this equation.
c) The half-life of the isotope Potassium 42 is 12.45hours. What percentage of the
isotope remains after 10 hours.
24. The temperature of a cake removed from an oven is 150º. Three minutes later its
temperature is 90º. How long will it take to cool to within 1º of the steady room
temperature of 20º.
25. The population of a town increases at a rate proportional to the number of people in
the town. In 1990 the town had a population of 32 908 and in 2000 the population was
43 185.
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90636
Estimate the
in 2008.
1– population
2
3
4
5
1
1– 1
2
3
4
26. The velocity of a runner over a period of three hours is given by
v  10  10 cos 2t , 0  t  3 (velocity in km/hour and time in hours)
Calculate the distance travelled by the runner in the first 2 hours.
27. Calculate the shaded area enclosed by the curve y 
y  0, x  0, x  3
5
4
3
2
1
– 1
y
4x  2
and the lines
x 1
1
– 1
2
3
4
x
2
4
6– 6
2
4
2
4
28. An
experimental cover for a garden is in the shape of half an ellipse rotated about the
y-axis.
x2 y 2
The equation for the ellipse is

1
36 16
Find the volume contained within the cover. Measurements are in metres.
y
4
2
1
3
2
1
2
1–
2
3
– 3
1
2
3
5

6
4
3
12
2
– 6
– 4
– 2
2
4
x
6
29. Find the area enclosed by the curve y  2 cos 3 x cos x and the x-axis from

x  0 to x 
6y
3
2
1
– 1




12
6
4
3
5
12

x
2
– 2
– 3
30. A car is timed from the moment it passes through an intersection. The velocity of the
car after this time is given by
12000
v(t ) 
0  t  50 t in seconds, v in m/sec
600  10t
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90636
5
4
3
2
1
5
How far from the intersection is4321 the car after 30 seconds?
Excellence Questions
5
1. The area enclosed between the graphs of
y  4 x  x 2 and y  3 is rotated about the
line y  3 . Find the volume formed.
y
1
2
3
4 x
2. Show that the half-life T of a radioactive substance is related to the decay constant k by
1
T  ln 2 , (where N  N 0 e  kt )
k
If the half-life of an isotope is 1600 years, how long does it take for one third of the
original amount to disintegrate.
3. A house was purchased in 1986 for $300 000 and sold in 1996 for $1.1 million. What
10
5
– 10
5 is the annual compounding percentage increase rate involved.
1– 5
2
1
2
3
4
4. Find the area of the closed loop of the curve y 2  x4  4  x  as shown in the diagram.
y
10
5
– 5
– 4
– 3
– 2
– 1
1
2
x
– 5
– 10
5. Solve the differential equation
Integration 2009
dy
 12e 2 x 6 y .
dx
11
90636
6. Find the total area enclosed by the
three curves shown.
The equations of the curves are:
A : y2  x
y
A
B
x
B : y  4( x  6)
2
C : y  x2  6x
C
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90636
Answers (numerical answers will vary depending on rounding used)
Answers: Achievement
12 x 5
u2
1. a)
b)
c) 2 x3  2 x 2  2 x  c
c
 eu  c
5
2
4
2
x 5x
x 5
x 3
d)
e) 8x + c
f)
g)

c
c
c
4
2
5
3
3
x 4 6 x 2
x1.5
x1.6
h)
i) 4x 2  c
j)
k)

c
2 x c
c
16 2
1.5
1.6
x 2 x3  x
l)
 
c
10 18 2
5
7t  3

p)
c
35
x4
 x2  c
4
2 x 4 3x 2
e)

c
4
4
e5 x
c
3. a)
5
e12 x
e)
c
12
2. a)
4. a) ln x  c
2 ln 1  x
4 x 0.6
m)
c
0.6
b)
5 x3
xc
3
3
c
n)
16 x 4
o)
c)
x4 x2
 c
4 2
d)
c)
2e 4  8 x
c
8
d)
 x  5
7
7
c
5 x 1 4 x 2

c
1
2
f) x  8 x  c
6e4 x
c
4
e 2 x e3 x

c
f)
2
3
b)
b) 5ln u  c
g) 4e x  c
c)
1
ln x  c
5
5e2 x
c
2
e2 x
h) x 
c
2
ln 4 x  3
c
d)
4
x 1
 ln x  c (these answers can be written in other ways)
1
1
 sin  5  6 x 
 cos 5 x
3sin 2 x
 x
c
c
c
5. a)
b)
c) 15cos    c
d)
5
2
6
3
 4 x 
 cos 

tan 8 x
cosec2 x
 3   c h) sin  x  3   c
c
c
e)
f)
g)


4
8
2
2 

3
6. a) 45
b) 13.5
c) 0.354
d) 4405.293
e) 12.91
f) 13.183
7. 45 1 3
8. 4
9. 3.225
10. 1
11. 3 112
12. 134.04
13. 965.1
14. y  6 x 2  8 x  10
15. y  2 x3  2 x 2  24
16. V  t 3  2t 2  12t  100
e)
c
f)
t2 t3
 , 66 2 3 m
2 60
2
19. s  sin  60t  50   1.781
3
20. y  4 x3  3x 2  cx  c
17. d 
Integration 2009
18. d  2t 3  30t
13
90636
21. a) y  3x 2  c
b) y 
x3
 x2  c
3
c) x 2  y 2  c
d) y 2  8 x  c
x2 y 2
  6x  2 y  c  0
2 2
24. 5 y  3e2 x  c
23. ln y  6ln x  c ( y  kx6 )
26. 4 y 2  x 2  252
27. y 
22.
28. 118m2
30. 1.685
25. y  e3x
31. 2.73m
34. 4885.8
Answers: Merit
 cos8 x  cos 4 x

c
1. a)
8
4
1  sin 2 x sin 4 x 

c) 
c
2 2
4 
3
 5
c
 3e
x
 25.83
29. 0.886
33. 100
x
2. a)
1
4
2
2
32. 0.230
sin
4
x
1

35. y 2 
4
2
sin10 x sin 4 x

c
10
4
3cos 2 x
c
d)
2
b)
5
c
c)  cos  x   c
c
5
1
sin 2 x 
3.  x 
c
2
2 
b) e
4. a) ln x 3  5  c
b) ln 3 x 2  4 x  1  c
3x2
2
d)
4  3x 2  8
5
5
c
c) 3ln 2 x 2  3  c
d) ln e x  12  c
5.  ln cos x  c
6. a) 4 x  14ln x  3  c
b) 6 x  13ln x  2  c
1.5
8  x  1
2
1  2x2   c
 8 x 1  c
7.
8. sin 1 x  c
9.

3
3
  x  12

 2  x  1  ln x  1   c
10. 2 
 2



1
11. 4.5
12.
13. 1.8856
14. 15
3
15. 1.6755
16. 632.1337
17. 407.15
18. 1.6442
19. Differentiate y twice and substitute.
 t2

20. y  e x
21. a) V  250   7t   6625
b) $500
2

1.5
22. y  20  5000e0.05t  50000  , 1164872insects
dN
 kN
b) N  N 0 e  kt
c) 57.3%
24. 23.2 minutes
dt
25. 53670 (answers for these questions will vary considerably depending on rounding) u
26. 23.78km
27. 9.227
28. 96
3 3
29.
30. 831.78m
8
23. a)
Integration 2009
14
90636
Answers: Excellence
1. 3.351
2. a)
4. 39.01
Integration 2009
b) 935.9 years
3. 13.87%
6y
2x
5. e  36e  c
15
6. 47.314
90636