11x1 t17 07 approximations

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

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Page 1: 11X1 T17 07 approximations

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

Page 2: 11X1 T17 07 approximations

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

Page 3: 11X1 T17 07 approximations

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

bfafab

A

2

Page 4: 11X1 T17 07 approximations

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafab

A

2

Page 5: 11X1 T17 07 approximations

Approximations To Areas (1) Trapezoidal Rule

y

x

y = f(x)

a b

y

x

y = f(x)

a b

bfafab

A

2

c

Page 6: 11X1 T17 07 approximations

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

Page 7: 11X1 T17 07 approximations

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

Page 8: 11X1 T17 07 approximations

y

x

y = f(x)

a b

Page 9: 11X1 T17 07 approximations

y

x

y = f(x)

a b d c

Page 10: 11X1 T17 07 approximations

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

Page 11: 11X1 T17 07 approximations

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

Page 12: 11X1 T17 07 approximations

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

In general;

Page 13: 11X1 T17 07 approximations

y

x

y = f(x)

a b d c

bfdfdb

dfcfcd

cfafac

A

2

22

bfdfcfafac

222

b

a

dxxfAreaIn general;

Page 14: 11X1 T17 07 approximations

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;

Page 15: 11X1 T17 07 approximations

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;

Page 16: 11X1 T17 07 approximations

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;

Page 17: 11X1 T17 07 approximations

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

Page 18: 11X1 T17 07 approximations

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

Page 19: 11X1 T17 07 approximations

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

Page 20: 11X1 T17 07 approximations

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

Page 21: 11X1 T17 07 approximations

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

Page 22: 11X1 T17 07 approximations

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

Page 23: 11X1 T17 07 approximations

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

Page 24: 11X1 T17 07 approximations

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

Page 25: 11X1 T17 07 approximations

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

Page 26: 11X1 T17 07 approximations

(2) Simpson’s Rule

Page 27: 11X1 T17 07 approximations

(2) Simpson’s Rule

b

a

dxxfArea

Page 28: 11X1 T17 07 approximations

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

Page 29: 11X1 T17 07 approximations

(2) Simpson’s Rule

b

a

dxxfArea

nevenodd yyyyh

243

0

intervals ofnumber

where

n

n

abh

Page 30: 11X1 T17 07 approximations

(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

Page 31: 11X1 T17 07 approximations

(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

Page 32: 11X1 T17 07 approximations

(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

Page 33: 11X1 T17 07 approximations

(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

Page 34: 11X1 T17 07 approximations

(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

Page 35: 11X1 T17 07 approximations

(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

Page 36: 11X1 T17 07 approximations

(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 %

Page 37: 11X1 T17 07 approximations

Alternative working out!!! (1) Trapezoidal Rule

Page 38: 11X1 T17 07 approximations

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

Page 39: 11X1 T17 07 approximations

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

Page 40: 11X1 T17 07 approximations

(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

Page 41: 11X1 T17 07 approximations

(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

Page 42: 11X1 T17 07 approximations

(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