11x1 t16 07 approximations (2011)

42
Approximations To Areas (1) Trapezoidal Rule y x y = f(x) a b

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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

bfafab

A

2

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafab

A

2

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafab

A

2

c

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafab

A

2

c

bfcfcb

cfafac

A

22

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafab

A

2

c

bfcfcb

cfafac

A

22

bfcfafac

22

y

x

y = f(x)

a b

y

x

y = f(x)

a b d c

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

In general;

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

b

a

dxxfAreaIn general;

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

b

a

dxxfArea

nothers yyyh

22

0

In general;

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

b

a

dxxfArea

nothers yyyh

22

0

s trapeziumofnumber

where

n

n

abh

In general;

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

NOTE: there is

always one more

function value

than interval

b

a

dxxfArea

nothers yyyh

22

0

s trapeziumofnumber

where

n

n

abh

In general;

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

5.0

4

02

n

abh

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

5.0

4

02

n

abh

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

5.0

4

02

n

abh

nothers yyyh

22

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

5.0

4

02

n

abh

nothers yyyh

22

Area 0

x 0 0.5 1 1.5 2

y 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

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

5.0

4

02

n

abh

nothers yyyh

22

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 2 2 2

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

5.0

4

02

n

abh

2units 996.2

03229.17321.19365.1222

5.0

nothers yyyh

22

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 2 2 2

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

5.0

4

02

n

abh

2units 996.2

03229.17321.19365.1222

5.0

πe exact valu

nothers yyyh

22

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 2 2 2

e.g.

points decimal 3 correct to

2 and 0between ,4 curve under the area

theestimate tointervals 4 with Rule lTrapezoida theUse

2

12 xxxy

5.0

4

02

n

abh

2units 996.2

03229.17321.19365.1222

5.0

πe exact valu

%6.4

100142.3

996.2142.3error %

nothers yyyh

22

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 2 2 2

(2) Simpson’s Rule

(2) Simpson’s Rule

b

a

dxxfArea

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

e.g. x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

e.g.

nevenodd yyyyh

243

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

e.g.

nevenodd yyyyh

243

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

e.g.

nevenodd yyyyh

243

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 4 4

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

e.g.

nevenodd yyyyh

243

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 4 2 4

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

e.g.

2units 084.3

07321.123229.19365.1423

5.0

nevenodd yyyyh

243

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 4 2 4

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

e.g.

2units 084.3

07321.123229.19365.1423

5.0

nevenodd yyyyh

243

Area 0

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 4 2 4

%8.1

100142.3

084.3142.3error %

Alternative working out!!! (1) Trapezoidal Rule

Alternative working out!!! (1) Trapezoidal Rule

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 2 2 2

Alternative working out!!! (1) Trapezoidal Rule

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 2 2 2

2 2 1.9365 1.7321 1.3229 0Area 2 0

1 2 2 2 1

22.996 units

(2) Simpson’s Rule

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 4 2 4

(2) Simpson’s Rule

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 4 2 4

2 4 1.9365 1.3229 2 1.7321 0Area 2 0

1 4 2 4 1

23.084 units

(2) Simpson’s Rule

x 0 0.5 1 1.5 2

y 2 1.9365 1.7321 1.3229 0

1 1 4 2 4

2 4 1.9365 1.3229 2 1.7321 0Area 2 0

1 4 2 4 1

23.084 units

Exercise 11I; odds

Exercise 11J; evens