VOLUMES OF REVOLUTION
Ø
Ø
Ø
Ø
Areas involving ! and " −axes
Volumes of revolution (! −axis)
Volumes of revolution (" −axis)
Volumes involving two curves
Warm Up
The graph shows the speed at
which an object travels over time.
a) Find the distance it travels in:
i.
1s
30 m
ii.
2s
60m
iii. 3s
90m
b) Find the area under the line
between:
i.
t = 0 and 1 30 units
ii.
t = 0 and 2 60 units
iii. t = 0 and 3 90 units
a) The speed is a constant 30 metres per second (m/s)
i. The object travels 30 metres in 1 s
ii. The object travels 60 metres in 2 s
iii. The object travels 90 metres in 3 s
b)
i. The area is 30 x 1 = 30 units2
ii. The area is 30 x 2 = 60 units2
iii. The area is 30 x 3 = 90 units2
The area gives the distances.
LINK
The line graph gives a rate of change.
The area under the curve gives the information about
the original data for this rate of change.
Thus , the area under a rate of change curve gives the
original data.
The original data is the primitive function of the curve.
Applications of Integration
Ø Distance, Velocity and Acceleration
Ø Volumes
Ø Physics - Work, Centre of Mass, Kinetic
Energy
Ø Probability
Ø Arc Length, Surface Area
THE DEFINITE INTEGRAL
The diagram below shows the region contained between a given curve ! =
#(%) and the % −axis, from % = ( to % = ), where ( ≤ ).
In the diagram, below, the region has been dissected into a number of thin
strips. Each strip is approximately a rectangle, but only roughly so, because
the upper boundary is curved. The area of the region is the sum of the area
of all the strips.
Let the width of the one of the strips be +%, where +% is thought as being
very small.
Area of strip~height × width
~# % +%
Adding up all the strips, using sigma notation for the sum.
$
Area of shaded region~ = (area of each strip)
!"#
$
~ = # % +%
!"#
INTEGRAL NOTATION
We can approximate the area under the curve using a large
number of rectangles by making the width of each rectangle
very small.
Taking an infinite number of rectangles, !" → 0
As !" → 0, the sum of rectangles → exact area
Area = lim - .(") !"
!"→$
= 1 .(") !"
NOTE:
The symbol ∫ is an old form of the letter S and is used to suggest
an infinite sum under a smooth curve (sum of rectangles).
We call ∫ "($) &$ an indefinite integral.
If we are finding the area under the curve " = $ % between % =
& and % = *, we write:
"
+ $(%) .%
!
"
We call ∫! $(%) .% a definite integral.
FUNDAMENTAL THEOREM OF
CALCULUS
The area enclosed by the curve ! = #(%), the % − axis
and the lines % = ( and % = ) is given by:
"
* # % +% = , ) − ,(()
!
Where ,(%) is the primitive function #(%)
Area and the Integral
Regions above the x-axis have p_________
Ositive signed area.
Regions below the x-axis have n_________ signed area.
egative
Suppose #(%) ≥ 0 in the domain % ∈ (, ) . Then
(
∫' "($) &$
yields the a________
between the curve and
re a
the _______ axis.
(
Calculating ∫' "($) &$
yields the s_________
are
igned
enclosed by the graph of ! = #(%) and the x-axis in the
domain % ∈ [____,
a ____
b ]
If #(%) is below the x-axis for % ∈ (, )
will be n__________.
egative
(
then ∫' "($) &$
Area Under A Curve
The diagram below shows the graph of " = $ % . Let 0, 2, 3 and 4
represent the positive area of their corresponding regions.
The regions with a positive signed area are __________
R and the
and
P
regions with a negative signed area are ______.
Q S
!
S
∫% "($) &$ will calculate 0 − 2 + 3_______.
The area of the region enclosed by $ % and the x-axis where % ∈
9, . is __________.
de or
feddel
I
The total area between the $ % and the x-axis is __________.
Iffce
ptQtrts
When finding the area of a region, you should produce a
s________
ketch of the region first. This is so then you can see fi there
are any “areas” b_______
ebwe the x-axis that you have to make positive.
However, if the function is clearly p_________
in the domain of
osibre
integration, then a s_________
is not necessary.
ketch
If the function is o_______
or e________,
use symmetry whenever
ven
dd
possible to make it easer to find the area.
QUESTION 1
By first sketching the following graph, find the area of
the region bounded by the curve and the x-axis.
! =2−%−%
E
t 2
E Ze te t 2
#
e
eat 2 trcetz
Ceti
l e
Getz
Getz
if
I
e
E de
2
Ee E E
2
4 2
1 1
I
f
5
Itf
7
80
units
QUESTION 2
a) Find the area of the region bounded by
$ % = % # − 8% $ + 15%
RE 8
f
1
15
eco 3 Ce 5
j
63g t
&
63 t
+ $(%) .%
12,52
ft
t
10031
373
81
2
632
f's
634 163
%
o
Is
E
I
213 132
b) Calculate
f
3 85 15e de
3
L
sN
18
3 8 4156de t
2572
4 853 151
85 15 de
14
10030
373
12,52
c) Explain briefly how your answer in (b) implies that the region
where % ∈ 0, 3 is larger than the region where % ∈ 3, 5
The
netsignedarea
more positive area
e
de is positive so theremust have been
than
negative area
1321
QUESTION 3
Consider the shaded region below:
Write down and evaluate the integral that represents the shaded
region by considering the region bounded by the curve and the
a Crore and
y p
Y
f
e
axis
y
or
y
y dy
a) " −axis
y
O
8
3g
b
12 units
16
Curveand e
axis
y p
b) % −axis
when
y
8
2 8
is
16
I6
8
8
V8
2
Je
de
ET
4025
4
12 units
QUESTION 4a
Find the area of the following region bounded by " = 2 ln % + 2
and both axes.
y
2 InCet2
y InCatz
Getz
e I eth
eY
Catz
et 2
e
I
e
e
k
e
2
2 dy
Ezek 2g
Re
26
20 263
26m43
244
2
2 244 I
2
2
2h22I
442
4h 2
2
QUESTION 4b
Find the area of the following shaded region:
" = tan'( % for % ∈ 0, 1
y
tan Ceti
etr
tang
tang I
e
glory
0
Ig
e dy
day
Choosel y
q
fantasy
flat
In 2
in 2
I
k
If
finccoso
o
fines
g
If
In 2 units
QUESTION 4c
Find the area of the following shaded region:
" = 3 sin'(
)
$
for % ∈ 0, 2
y 3 sun E
Ate
2
y
y
y
y
3
3 sit
355 CD
3 52 342
E
n
F site
sun
8
13
25th
31
342
Jang dy
2
o
6
34
cosy
34
cos
34
6cost
605
353
6
0 6
3T1
31 6 units
z
QUESTION 5
By drawing a sketch and shading the appropriate region, evaluate
the following integral.
$
+ ln % .%
(
2
42
e
I
s is
n
d
4
y Ine
Leine
e
e
eY
Exinz
242
Jey dy
Ey
eh
21h2
2
242
242
p
I
so
1
i
AREA UNDER TWO CURVES
If $ % and D(%) are two continuous functions such that $(%) ≥
D(%) for all % ∈ &, * , then the area of the region bounded by the
two areas in the domain % ∈ &, * is given by:
"
fld g
F = + … … … … … e… . .%
!
If you are asked to find the area enclosed by two curves, without
being given any bounds, it means that you are required to first find
the points of i___________
of
intersection , and those will be your l_________
in its
integration.
!"
− @ # % +%
!!
)"
)#
)"
)$
+ D % .% − + $ % .% − + $ % .% + + $ % .%
)!
)!
)$
)"
)#
)"
+ D % .% − + $ % .%
)!
)!
)"
+ D % − $ % .%
)!
A + −C = A − −C
A+C =A+C
QUESTION 6
7C
Mo
09
15,8
&
The diagram below shows a sketch of ' = 8$ − 7 − $ and
' = $ & − 4x + 3
a)
64
Find their point of intersection.
8
7
5
25 12 410 0
E Get 5 0
5
x
5
4
where 1
Where S
CX 17 0
r0
does
e
3
y
y
867 7
865 7
45 0
65 8
and 5,8
b) Hence, find the area of the region between the two
curves.
8
7
5
624
3
de
r
J
2 712 10 de
Ets
we
50
253 150
33 6
250 100 33 th
104
0
2431
64
a
2
Y
units
QUESTION 7
The curves " = % and % = 8" − " $ are drawn below.
a) Find where the curves intersect.
b) Find the area of region A.
c) Find the area of region B.
Region
y y y0
y
71
yCy 77
At 1 0 e
only 7
b
O or 7
y
0
I
90
a
Region A
Ex
7
7
t
Foy
4ft ay I
Rft 48551
41 t
52
60
161
196 343
16g
a
28 t
y dy
44573
60 5131343
492
41
256
1889
j
0
units
B
fly
9
y
161
185 833
161
256 533 161
343 or 571units
had
QUESTION 8
The curves % = " + 5 and % = −" $ − 6" − 5 are drawn below.
Find the area of the shaded region.
yes y
2
Cytz Cyt5
y
y
I
gg
19
f
5
6 1235
5
17
50
0
or
y
Cyts dy
2
14 20
I
2
y ioay
I IF
units
Gy 5
5
10 0
y't74
I
by
so
5
HOMEWORK
SEE WORKSHEET AND FOR EXTRA PRACTICE ON INTEGRATION
(EX 5G)