Lesson 3 Definite Integral

8/3/2019 Lesson 3 Definite Integral

1/23

ANTIDERIVATIVES

(INTEGRAL)


2/23

THE DEFINITE INTEGRAL

define and interpret definite integral,

identify and distinguish the different properties of

the definite integrals; and

evaluate definite integrals

OBJECTIVES:


3/23

IfF(x) is the integral off(x)dx, that is, F(x) = f(x)dx

and ifa and b are constants, then the definiteintegral is:

)a(F)b(F

xFdx)x(fb

a

b

a

where a and b are called lower and upper limits of

integration, respectively.

The definite integral link the concept of area to

other important concepts such as length, volume,

density, probability, and other work.

THE DEFINITE INTEGRAL


4/23

baervalclosedtheindefinedisxfprovided

dxxfdxxf

thenbaIf

b

a

a

b

,int)(

)()(

,.1

.)()()(

),(int)(.2

existsbfandafprovideddxxf

thenxfofegraltheisxFandbaIf

b

a

PROPERTIES OF DEFINITE INTEGRAL

.0)()()()()()(

,

aFaFCaFCaFCxFdxxfisThat

b

a

b

a


5/23

dxxfdxxfdxxfdxxfxfxf nb

a

b

a

b

an )(......)()()()....()(.3 2121

b

c

b

a

c

a dxxfxfdxxf

thenbcawhere

baervalclosedtheinfunctioncontinuousisxfIf

)()()(

,,

,.int)(.4


6/23

To obtain the definite integral of a function,

evaluate first its indefinite integral. Then applying

the limits of integration, that is, substitute the

upper limit of integration to all the variables

contained in the indefinite integral, minus the

function value of the indefinite integral using thelower limit of integration.


7/23

1

1

2.1 dxx

EXAMPLE:

1

0

3 10)dx.2x(4x.2

9

1.3 )dy.yy(3


8/23

EXERCISES:

1

0

)23(.1 dxx

2

3

2)1(.2 dyyy

a

dtta0

2

.3

5

2

)57)(25(.4 dxxx

3

1

4

2

2.5 dmm

m

2/1

2/1

7)13(.6 dxx

2

0

32 1.7 dyyy


9/23

INTEGRATION OF ABSOLUTE VALUE FUNCTION

0xif

0xifxRe

x

xcall

EXAMPLE

dxx.14

2

1082

02

)4(

2

)2(0

22

x

224

0

20

2

2

4

0

0

2

4

2

xx

xdxxdxdx1st solution

0xif

0xifx

x

x

0 1-2 32 4-1

0x 0x


10/23

2nd solution

(4,4)4;(4)f

(0,0)0;(0)f

,2)(-22;f(-2)

y)(x,

xf(x)

let

-1 1-2 432

(4,4)

-2,2

0

8)4)(4(2

1

2)2)(2(21

2

1

A

A

10

82

21

4

2

AAdxx


11/23

3

31.2 dxx

1

st

solution

1x0,x-1if1

1x0,x-1if11

x

xx

102

20

2

1

2

3

2

15

2

1

22

x-x

111

3

1

21

3

2

3

1

1

3

3

3

x

x

dxxdxxdxx


12/23

2nd solution

(3,2)2;(3)f

(1,0)0;(1)f

,4)(-34;f(-3)

y)(x,

x-1f(x)

let

-1 1-2-3 32

(3,2)

-3,4

0

2)2)(2(2

1

8)4)(4(2

1

2

1

A

A

10

28

1 213

3

AAdxx


13/23

2

1

3.3 dxxx

0xif

0if

33

33

3

xxx

xxxxxx

011

01

0

2

3

xxx

xx

xx

011 01

0

2

3

xxx

xx

xx

,11,0-:SS 1,0,-1-:SS

4

11

2

13

1

03

2

1

0

133

dxxxdxxxdxxxdxxx


14/23

2553x.5

732.4

5

0

7

1

dx

dxx


15/23

INTEGRATION OF PIECEWISE FUNCTION

EXAMPLE

4x0,2

2

1

0x2-,2

f(x);)(.1

2

4

2 x

x

dxxf

solution

3

56x2

4

x

3

xx2

dx2x2

1dxx2dx)x(f

4

0

20

2

3

0

2

4

0

24

2


16/23

INTEGRATION OF ODD AND EVEN FUNCTIONS

x-integers,oddFor;x-integers,evenFor

:Re

nn

nn

xx

call

fofdomainxallforf(x)f(-x)ifevenbetosaidis Function

fofdomainxallforf(x)f(-x)ifoddbetosaidis Function

The graph of an even function is symmetric about the y-axis.

The graph of an odd function is symmetric about the origin.


17/23

Theorem:1. Iffis odd on [-a,a] then 0)(

a

adxxf

a

-a

1R

2R

0

)()()(

21

0

0

RofareaRofarea

dxxfdxxfdxxf

a

a

a

a


18/23

Theorem:2. Iffis even on [-a,a] then

aa

adxxfdxxf

0)(2)(

a-a

1R

a

a

a a

a

dxxf

RofareaRofarea

dxxfdxxfdxxf

0

2

21

0

0

)(2

Rofarea2

)()()(

2R


19/23

oddisfunctionthebecause01.1

3

3 2

3

dttt

EXAMPLE


20/23

6

4

2

2

6

0

0

2-

)(.)(.

f(x)dx.)(.

0,2

0,2)(

given thatintegraltheevaluatepart,eachIn.1

dxxfddxxfb

cdxxfa

xifx

xifxxf

EXERCISES


21/23

5

21

1

1

10

1

1

0

)(.)(.

f(x)dx.)(.

1,21,2)(

given thatintegraltheevaluatepart,eachIn.2

dxxfddxxfb

cdxxfa

xifxifxxf

EXERCISES


22/23

a b dc

Problem 19 page 360

Area =0.8

Area =2.6

Area =1.5

d

c

a

b

a

)(.)(.

f(x)dx.)(.

findtofigurein theshownareastheUse

a

c

bdxxfddxxfb

cdxxfa


23/23

Answers : Problem 19 page 360

0.8dx)x(f.ab

a

2.6dx)x(f.bc

b

-0.32.6-1.50.8dx)x(f.dd

a

1.8-2.6-0.8f(x)dx.cc

a

Lesson 3 Definite Integral

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