11x1 t14 07 approximations

Approximations To Areas(1) Trapezoidal Rule

y

x

y = f(x)

a b

Approximations To Areas(1) Trapezoidal Rule

y

x

y = f(x)

a b

Approximations To Areas(1) Trapezoidal Rule

y

x

y = f(x)

a b

bfafabA

2

Approximations To Areas(1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafabA

2

Approximations To Areas(1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafabA

2

c

Approximations To Areas(1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafabA

2

c

bfcfcbcfafacA

22

Approximations To Areas(1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafabA

2

c

bfcfcbcfafacA

22

bfcfafac

2

2

y

x

y = f(x)

a b

y

x

y = f(x)

a bdc

y

x

y = f(x)

a bdc

bfdfdb

dfcfcdcfafacA

2

22

y

x

y = f(x)

a bdc

bfdfdb

dfcfcdcfafacA

2

22

bfdfcfafac

22

2

y

x

y = f(x)

a bdc

bfdfdb

dfcfcdcfafacA

2

22

bfdfcfafac

22

2

In general;

y

x

y = f(x)

a bdc

bfdfdb

dfcfcdcfafacA

2

22

bfdfcfafac

22

2

b

a

dxxfAreaIn general;

y

x

y = f(x)

a bdc

bfdfdb

dfcfcdcfafacA

2

22

bfdfcfafac

22

2

b

a

dxxfArea

nothers yyyh 2

2 0

In general;

y

x

y = f(x)

a bdc

bfdfdb

dfcfcdcfafacA

2

22

bfdfcfafac

22

2

b

a

dxxfArea

nothers yyyh 2

2 0

s trapeziumofnumber

where

nn

abh

In general;

y

x

y = f(x)

a bdc

bfdfdb

dfcfcdcfafacA

2

22

bfdfcfafac

22

2

NOTE: there is always one more function value than interval

b

a

dxxfArea

nothers yyyh 2

2 0

s trapeziumofnumber

where

nn

abh

In general;

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

5.04

02

nabh

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

5.04

02

nabh

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

5.04

02

nabh

nothers yyyh 2

2Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

5.04

02

nabh

nothers yyyh 2

2Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 1

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

5.04

02

nabh

nothers yyyh 2

2Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 12 2 2

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

5.04

02

nabh

2units996.2

03229.17321.19365.12225.0

nothers yyyh 2

2Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 12 2 2

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

5.04

02

nabh

2units996.2

03229.17321.19365.12225.0

πe exact valu

nothers yyyh 2

2Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 12 2 2

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimatetointervals4with RulelTrapezoida theUse

21

2 xxxy

5.04

02

nabh

2units996.2

03229.17321.19365.12225.0

πe exact valu

%6.4

100142.3

996.2142.3error %

nothers yyyh 2

2Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 12 2 2

(2) Simpson’s Rule

(2) Simpson’s Rule

b

a

dxxfArea

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

intervalsofnumber

where

nn

abh

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

intervalsofnumber

where

nn

abh

e.g.x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

intervalsofnumber

where

nn

abh

e.g.

nevenodd yyyyh 24

3Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

intervalsofnumber

where

nn

abh

e.g.

nevenodd yyyyh 24

3Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 1

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

intervalsofnumber

where

nn

abh

e.g.

nevenodd yyyyh 24

3Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 14 4

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

intervalsofnumber

where

nn

abh

e.g.

nevenodd yyyyh 24

3Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 14 2 4

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

intervalsofnumber

where

nn

abh

e.g.

2units 084.3

07321.123229.19365.14235.0

nevenodd yyyyh 24

3Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 14 2 4

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh 24

3 0

intervalsofnumber

where

nn

abh

e.g.

2units 084.3

07321.123229.19365.14235.0

nevenodd yyyyh 24

3Area 0

x 0 0.5 1 1.5 2y 2 1.9365 1.7321 1.3229 0

1 14 2 4

%8.1

100142.3

084.3142.3error %

Exercise 11I; odds

Exercise 11J; evens