Topic 3 Introduction to Calculus - Integration
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Transcript of Topic 3 Introduction to Calculus - Integration
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7/28/2019 Topic 3 Introduction to Calculus - Integration
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TOPIC 3 INTRODUCTION TO
CALCULUS [MO1,MO2,MO3]Integration
Application of Integration
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Definition
Integration or the process of finding the antideravative
is one of the most important operations in calculus
It is opposite process of the differentiation
The notation will denote any anti-
derivative of .
is called as the integral of
is called the integrand
dxxf )()(xf
)(xf
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Types of integration
Indefiniteintegral
Definiteintegral
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Indefinite integral
If is the derivative of the function ,then is called the antideravative or
indefinite integral of .
We write integral of
)(xf )(xF)(xF
)(xf
cxFdxxf )()(
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Basic rules in integration(Indefinite integral)
dxxfcdxxfc )()(
dxxgdxxfdxxgxf )()()]()([
dxxgdxxfdxxgxf )()()]()([
dxxgbdxxfadxxgbxfa )()()]()([
Constant multiple rule
Sum rule
Difference rule
Linearity rule
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cdx 00
1,1
1
ncnx
dxxn
n
Constant rule
Power rule
Power rule
1 ln | |x dx x c
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Example 1 Find the following antiderivatives:
cxxx
cxxx
dxdxxdxx
dxxx
7
6
712
315
73
)73(
36
1215
25
25Solution:
dxxx
)73( 25
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EXAMPLE 2
Find the integral of(
) .
Solution:
= ( )
=
+
+
+
=
=
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EXAMPLE 3
Integrate +
.
Solution:
=
= 4 5
= +
4 +
=
4
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Integrating
=+
+
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Example 4
Find the 7 1 0 .
Solution: =
+
+
=+
9
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Example 5
Integrate 2 5
Solution: = 5 2
=+
+
= +
9
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Integration of and +
= ln ||
(
+ ) =
ln
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Example 6
Integrate the following functions with respect tox:
= 3 l n
+ =
l n | 2 5 |
=
ln 3 2
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Method of integration
Integration by using substitution:
=
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Example 7 Exaluate 2 1 . Solution:
= 1 2
= 1
= 2.
:
1 2 = =
=
=+
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Exercises
1
+ 2
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DEFINITE INTEGRALIf F(x) is any indefinite integral of f(x) so that F(x) = f(x) then:
( ) [ ( )] ( ) ( )b
b
aa
f x dx F x F b F a
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Given a function f(x) of a real variable
xand an interval [a, b] of the real line,
the integral
is equal to the area of a region in the
xy-plane bounded by the graph off,
the x-axis, and the vertical lines x= aand x= b, with areas below the x-axis
being subtracted.
b
adxxf )(
http://upload.wikimedia.org/wikipedia/commons/4/42/Integral_example.png -
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Example 8
2
0
2
)2( dxxx
Evaluate:
(a)
(b)
4
1 2
1dx
xx
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Example 9 Find the following antiderivatives:
(a)
(b)
dxxx 72 )32(4
dxxx 2
1
43 5
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Application of integrationArea Under a Curve Lesson Outcome:
Apply the definite integrals to find the area under curves.
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The definite integral can be used to find the area between a graph
curve and the x axis, between two given x values. This area is
called the area under the curve regardless of whether it is above
or below the x axis.Area when the curve is above the x-axis
The area of the region that lies
under the curve y = f(x), and
the linesx = a
,x = b
, wheref
is
continuous and f(x) 0 for allx
in [a,b] is
b
adxxf )(Area
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Area when the curve is below the x-axis
The area of the region that lies under the curve y = f(x), and the linesx = a,x = b, wheref is continuous and f(x) 0 for allx in [a,b] is
b
adxxf )(Area
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Example 10MTH1022
Find the area bounded by ,22 xy
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2 3 4
x = 0 x = 1
thex-axis
and the linesx = 0 and x = 1.
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SolutionMTH1022
Given that the function is and the linesx = 0 andx = 122
xy
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2 3 4
x = 0 x = 1
22 xy
= 0 = 1
=
=
=
=
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Example 11MTH1022
Find the area bounded by the lines and the
x-axis.4,3,2 xxxy
3x 4x
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7
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solutionMTH1022
3x 4x
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7
= 2
= 3 = 4
= 2
2 =
2 4
3
=
2 4
2 3
=
8
9
6
= 0
=
. 3
2
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Example 12
MTH1022
Find the area underneath the curve from x = -1 to x = 1.3
xy
x = -1 x = 1
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
BECAREFUL OF THE INTERVAL VALUE!!
FOR INTERVAL THAT PASSES 0IT IS
ADVICED THAT YOU SHOULDSEPARATE THEM IN 2 INTEGRAL
EQUATIONS.
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Solution
MTH1022
Find the area underneath the curve from x = -1 to x = 1.3
xy
x = -1 x = 1
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
= =
= =
=
=
01
+
10
=
= 0
0
=
=
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Example 13MTH1022
Find the area bounded by the lines , they-axis and .1,0 yy 2xy
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
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SolutionMTH1022
Find the area bounded by the lines , they-axis and .1,0 yy 2xy
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
= = =
= 0 = 1
=
10
=
=
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Example 14MTH1022
Find the area bounded by , the lines y = 2, y = 4 and the
y-axis.x
y2
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5
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SolutionMTH1022
Find the area bounded by , the lines y = 2, y = 4 and the
y-axis.x
y2
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5
= =
= 2 = 4
= [ln ||] 42
= (ln 4) (ln 2)= 2.771 1.386= 1.385
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Exercise
1. Find the area bounded by the graph of y=x3and x-axis from x=-1 to x=1.
2. Find the area bounded by the graph of y=x3and y-axis from y=1 to y=4.
MTH1022
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Solutions1. Given that = and = 1 = 1 .
Point x will form:
= 1 = 0 & = 0 = 1
=
=
01
+
10
=
=
=
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Solution2. Given that y=x3 and = 1 = 4 .
= = =
=
+
41
=
4
1 =
4
1
=
= 4.012
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Area Under a Curve
In order to find area under a curve/graph,you must first sketch the curve/graph todetermine the actual region.
Area under a curve is a region formedbetween the curve andx-axis; can be eitherabove or below thex-axis.
We need to take the absolute value to find
the area.
MTH1022
Summary
Area Under
A Curve
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Area between two curves at the x-axis
Iff and g are continuous on the
interval [a,b], and iff(x)g(x) for all
x in [a,b], then the area of the region
bounded byy = f(x), y = g(x),x = a, and x = b is:
b
adxxgxf )]()([A
Area Between Two CurvesMTH1022
To remember the above formula we write
b
adxfunctionlower-functionupperA where .bxa
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Area between two curves at the y-axis
Iff(y) and g(y) are continuous on the
interval [c,d], and iff(y) g(y) for ally
in [c,d], then the area of the region
bounded byx = f(y), x = g(y), y = c,
and y = d is
d
cdyyvyu )]()([A
Area Between Two CurvesMTH1022
To remember the above formula we write
d
cdyfunctionleft-functionrightA where .dyc
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Find the area of the region bounded by the curve and the
linesy = - x, x = 0 and x = 1.
22 xy
Example 15
MTH1022
Answer:2
unit6
52A
-4
-2
0
2
4
6
8
10
12
-4 -3 -2 -1 0 1 2 3 4
x = 1x = 0
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xy
Solution
MTH1022
-4
-2
0
2
4
6
8
10
12
-4 -3 -2 -1 0 1 2 3 4
x = 1x = 0
22 xy
UF
LF
= 2
= 2
=
2
1
0=
2 1
0
=
= 2
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Example 16MTH1022
Find the area of the region that is enclosed between the curveand the line .
2xy
6 xy
Answer:2
unit6
5
20A
0
1
2
3
4
5
6
7
8
9
10
-4 -3 -2 -1 0 1 2 3 4
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Example 17MTH1022
Find the area of the region bounded by the curves and2
4 xy
Answer: 2unit9Axxy 2
2
-10
-5
0
5
10
15
20
-4 -3 -2 -1 0 1 2 3 4
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Example 18MTH1022
Find the area of the region enclosed by and y = x2 ,
integrating with respect toy.
2yx
Answer:2
unit2
14A
-4
-3
-2
-1
0
1
2
3
4
-2 0 2 4 6 8 10
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Example 19MTH1022
Find the area between the curve and .102 yx
2)2( yx
Answer:
-4
-3
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30
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Example 20MTH1022
Determine the area of the region bounded by and.
2yx
Answer: 2unit3
22A
.22
yx
-4
-3
-2
-1
0
1
2
3
4
-10 -5 0 5 10
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A B t T C
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Summary:
Area Between Two CurvesMTH1022
1. Sketch the graph by the given functions.
2. Identify the region bounded by the
functions or curves.
3. Determine the curves intersection or
determine the limits of integration.
4. Integrate the function on related axis.
Steps to find the area between two curves:
Area
Between
Two Curves