Midpoint ApproximationTrapezoidal RuleErrorSimpson’s Rule ...

[Pages:44]Midpoint Approximation Trapezoidal Rule Error Simpson's Rule

Midpoint Approximation

Sometimes,

we

need

to

approximate

an

integral

of

the

form

Rb

a

f (x)dx

and

we

cannot find an antiderivative in order to evaluate the integral. Also we may

need

to

evaluate

Rb

a

f (x)dx

where

we

do

not

have

a

formula

for

f (x)

but

we

have data describing a set of values of the function.

Review

We might approximate the given integral using a Riemann sum. Already we

have looked at the left end-point approximation and the right end point

approximation

to

Rb

a

f

(x

)dx

in

Calculus

1.

We

also

looked

at

the

midpoint

approximation M:

Midpoint Rule If f is integrable on [a, b], then

Zb

n

X

f (x)dx Mn = f (x?i )x = x(f (x?1) + f (x?2) + ? ? ? + f (x?n)),

a

i =1

where

b-a

1

x = n

and xi = a+i x and x?i = 2 (xi-1+xi ) = midpoint of [xi-1, xi ].

Annette Pilkington

Approximating an integral

Midpoint Approximation Trapezoidal Rule Error Simpson's Rule

Midpoint Approximation

Example

Use

the

midpoint

rule

with

n

=

6

to

approximate

R4

1

1 x

dx

.

(= ln(4) = 1.386294361)

Fill in the tables below:

Annette Pilkington

Approximating an integral

Midpoint Approximation Trapezoidal Rule Error Simpson's Rule

Midpoint Approximation

Example

Use

the

midpoint

rule

with

n

=

6

to

approximate

R4

1

1 x

dx

.

(= ln(4) = 1.386294361)

Fill in the tables below:

x

=

4-1 6

=

1 2

Annette Pilkington

Approximating an integral

Midpoint Approximation Trapezoidal Rule Error Simpson's Rule

Midpoint Approximation

Example

Use

the

midpoint

rule

with

n

=

6

to

approximate

R4

1

1 x

dx

.

(= ln(4) = 1.386294361)

Fill in the tables below:

x

=

4-1 6

=

1 2

xi x0 = 1 x1 = 3/2 x2 = 2 x3 = 5/2 x4 = 3 x5 = 7/2 x6 = 4

Annette Pilkington

Approximating an integral

Midpoint Approximation Trapezoidal Rule Error Simpson's Rule

Midpoint Approximation

Example

Use

the

midpoint

rule

with

n

=

6

to

approximate

R4

1

1 x

dx

.

(= ln(4) = 1.386294361)

Fill in the tables below:

x

=

4-1 6

=

1 2

xi x0 = 1 x1 = 3/2 x2 = 2 x3 = 5/2 x4 = 3 x5 = 7/2 x6 = 4

x?i

=

1 2

(xi

-1

+ xi )

f (x?i ) =

1 x?i

x?1 = 5/4 4/5

x?2 = 7/4 4/7

x?3 = 9/4 4/9

x?4 = 11/4 4/11

x?5 = 13/4 4/13

x?6 = 15/4 4/15

Annette Pilkington

Approximating an integral

Midpoint Approximation Trapezoidal Rule Error Simpson's Rule

Midpoint Approximation

Example

Use

the

midpoint

rule

with

n

=

6

to

approximate

R4

1

1 x

dx

.

(= ln(4) = 1.386294361)

Fill in the tables below:

x

=

4-1 6

=

1 2

xi x0 = 1 x1 = 3/2 x2 = 2 x3 = 5/2 x4 = 3 x5 = 7/2 x6 = 4

x?i

=

1 2

(xi

-1

+ xi )

f (x?i ) =

1 x?i

x?1 = 5/4 4/5

x?2 = 7/4 4/7

x?3 = 9/4 4/9

x?4 = 11/4 4/11

x?5 = 13/4 4/13

x?6 = 15/4 4/15

M6

=

P6

1

f

(x?i

)x

=

1^4

25

+

4 7

+

4 9

+

4 11

+

4 13

+

4~

15

=

1.376934177

Annette Pilkington

Approximating an integral

Midpoint Approximation Trapezoidal Rule Error Simpson's Rule

Trapezoidal Rule

We

can

also

approximate

a

definite

integral

Rb

a

f

(x

)dx

using

an

approximation

by trapezoids as shown in the picture below for f (x) 0

The

area

of

the

trapezoid

above

the

interval

[xi ,

xi +1 ]

is

x

h

(f

(xi

)+f 2

(xi

+1

)

i .

Trapezoidal Rule If f is integrable on [a, b], then

Zb

x

a f (x)dx Tn = 2 (f (x0) + 2f (x1) + 2f (x2) + ? ? ? + +2f (xn-1) + f (xn))

where

x = b - a and n

Annette Pilkington

xi = a + i x and.

Approximating an integral

Midpoint Approximation Trapezoidal Rule Error Simpson's Rule

Trapezoidal Rule

where

Zb

x

f (x)dx Tn = a

(f (x0) + 2f (x1 ) + 2f (x2 ) + ? ? ? + +2f (xn-1 ) + f (xn )) 2

b-a x =

n

and xi = a + ix and.

Example

Use

the

trapezoidal

rule

with

n

=

6

to

approximate

R4

1

1 x

dx

.

(=

ln(4) = 1.386294361)

Annette Pilkington

Approximating an integral

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