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Announcements

Topics:

Sec1ons6.3(definiteintegral+area)and6.4(FTC)*Readthesesec1onsandstudysolvedexamplesinyour

textbook!

WorkOn:

Assignments9,20,andpartof2 Exercisesinthetextbookasassignedinthecoursepack

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Area

Howdowecalculatethe

areaofsomeirregular

shape?

Forexample,howdowe

calculatetheareaunder

thegraphoffon[a,b]?

Y

A B

F

T

Area = ?

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Area

Approach:

T

Y

A B

F

x0=

x1

x2

x3

= x4

numberof

rectangles:

widthofeach

rectangle:

Weapproximatetheareausingrectangles.

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Area

Le#handes+mate:

Area f(x0

)x + f(x1

)x + f(x2

)x + f(x3

)x

(f(x0) + f(x1) + f(x2)+ f(x3))x

f(x i)x

i= 0

3

RiemannSum

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Area

Righthandes+mate:Lettheheightofeachrectanglebegivenbythevalue

ofthefunc1onattherightendpointoftheinterval.

T

Y

A B

F

x0= x

1x2

x3 = x4

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Area

Righthandes+mate:

Area f(x1

)x + f(x2

)x + f(x3

)x + f(x4

)x

(f(x1) + f(x2)+ f(x3) + f(x4 ))x

f(x i)x

i=1

4

RiemannSum

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Area

Midpointes+mate:Lettheheightofeachrectanglebegivenbythevalue

ofthefunc1onatthemidpointoftheinterval.

T

Y

A B

F

x1

x2

x3

x4

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Area

Midpointes+mate:

Area f(x1

*)x + f(x

2

*)x + f(x

3

*)x + f(x

4

*)x

f(x i*)x

i=1

4

RiemannSum

(f(x1*) + f(x2

*) + f(x3

*) + f(x4

*))x

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RiemannSumsandthe

DefiniteIntegralDefini+on:

Thedefiniteintegralofafunc1ononthe

intervalfromatobisdefinedasalimitoftheRiemannsum

whereissomesamplepointintheinterval

and

f(x)dx = limn

f(x i*)x

i=1

n

a

b

f

x =b a

n.

xi

*

[xi1,

xi]

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Es1ma1ngUsingLeYEndpoints

-1 0 1 2 3 4 5 6 7 8

2.5

5

.

(0.5t+ 2)dt

0

5

L5 =

f(0) 1+ f(1) 1+ f(2) 1+ f(3) 1+ f(4) 1

= 2+ 2.5+ 3+ 3.5+ 4

=15

t=5 0

5=1

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Es1ma1ngUsingRightEndpoints

-1 0 1 2 3 4 5 6 7 8

2.5

5

.

(0.5t+ 2)dt

0

5

R5 = f(1) 1+ f(2) 1+ f(3) 1+ f(4) 1+ f(5) 1

= 2.5+ 3+ 3.5+ 4 + 4.5

=17.5

t=5 0

5=1

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Es1ma1ngUsingMidpoints

-1 0 1 2 3 4 5 6 7 8

2.5

5

.

(0.5t+ 2)dt

0

5

M5 = f(0.5) 1+ f(1.5) 1+ f(2.5) 1+ f(3.5) 1+ f(4.5) 1

= 2.25+ 2.75+ 3.25+ 3.75+ 4.25

=

16.25

t=5 0

5=1

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et2

dt

0

1

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25

0.25

0.5

0.75

1

.

Es1ma1ngUsingLeYEndpoints

L4 = (f(0)+ f(0.25) + f(0.5) + f(0.75)) 0.25

= (e0

2

+ e0.25

2

+ e0.5

2

+ e0.75

2

) 0.25

0.822

t=1 0

4= 0.25

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et2

dt

0

1

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25

0.25

0.5

0.75

1

.

Es1ma1ngUsingRightEndpoints

R4= (f(0.25)+ f(0.5)+ f(0.75)+ f(1)) 0.25

= (e0.25

2

+ e0.5

2

+ e0.75

2

+ e1

2

) 0.25

0.664

t=1 0

4= 0.25

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et2

dt

0

1

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25

0.25

0.5

0.75

1

.

Es1ma1ngUsingMidpoints

M4 = (f(0.125)+ f(0.375)+ f(0.625)+ f(0.875)) 0.25

= (e0.125

2

+ e0.375

2

+ e0.625

2

+ e0.875

2

) 0.25

0.749

t=1 0

4= 0.25

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TypesofIntegrals

IndefiniteIntegral

DefiniteIntegral

f(x)dx=

F(x)+

C an1deriva1veoff

func1onofx

f(x)dx = net areaa

b

number

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TheDefiniteIntegral

Interpreta+on:

If,thenthedefinite

integralistheareaunderthecurvefrom

atob.

Y

A B

F

T

f 0

Area = f(x)dxa

b

y = f(x)

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TheDefiniteIntegral

Interpreta+on:

Ifisbothposi1veand

nega1ve,thenthedefiniteintegralrepresentsthe

NETorSIGNEDarea,i.e.

theareaabovethexaxis

andbelowthegraphoffminustheareabelowthe

xaxisandabovef

f

f(x)dx1

4

= net area

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DefiniteIntegralsandArea

Example:

Evaluatethefollowingintegralsbyinterpre1ng

eachintermsofarea.

(a) (b)

(c)

(x 1) dx0

3

1 x 2 dx0

1

sinx dx

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Proper1esofIntegrals

Assumethatf(x)andg(x)arecon1nuous

func1onsanda,b,andcarerealnumberssuch

thata

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Proper1esofIntegrals

Assumethatf(x)andg(x)arecon1nuous

func1onsanda,b,andcarerealnumberssuch

thata

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Summa1onPropertyofthe

DefiniteIntegral(6)Supposef(x)iscon1nuousontheinterval

fromatobandthat

Then

a c b.

f(x)dxa

b

= f(x)dxa

c

+ f(x)dxc

b

.

Y

A B

F

T

C

f(x)dxa

c

f(x)dxc

b

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TypesofIntegrals

IndefiniteIntegral

DefiniteIntegral

f(x)dx=

F(x)+

C an1deriva1veoff

func1onofx

f(x)dx = net areaa

b

number

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TheFundamentalTheoremofCalculus

Ifiscon5nuousonthen

whereisanyan1deriva1veof,i.e.,

f(x)dxa

b

= F(x) ab = F(b) F(a)

f

[a, b],

F

f

F'=

f.

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Differen1a1onandIntegra1on

asInverseProcessesIffisintegratedandthendifferen1ated,we

arrivebackattheoriginalfunc1onf.

IfFisdifferen1atedandthenintegrated,wearrivebackattheoriginalfunc1onF.

d

dxF(x)dx = F(x)

a

b

a

b

d

dxf(t)dt

a

x

= f(x)

FTCII

FTCI

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Applica1on

Example:

Supposethatthegrowthrateofafishisgivenby

thedifferen1alequa1on

wheretismeasuredinyearsandLismeasuredincen1metresandthefishwas0.0cmatage

t=0(1memeasuredfromfer1liza1on).

dL

dt= 6.48e

0.09t

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