topic 3 introduction to calculus - integration

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


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:


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

topic 3 introduction to calculus - integration

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