Numerical Integration

Numerical Integration

Mohammad Tawfik #WikiCourses

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Mohammad Tawfik




Objectives

• The student should be able to

– Understand the need for numerical integration

– Derive the trapezoidal rule using geometric

insight

– Apply the trapezoidal rule

– Apply Simpson’s rule




Need for Numerical Integration!

6

1101

2

1

3

1

231

1

0

231

0

2

x

xxdxxxI

11

0

1

0

1 eedxeI xx

1

0

2

dxeI x




Area under the graph!

• Definite integrations always result in the

area under the graph (in x-y plane)

• Are we capable of evaluating an

approximate value for the area?




Example

• To perform the

definite integration of

the function between

(x0 & x1), we may

assume that the area

is equal to that of the

trapezium:

0101

2

1

0

xxyy

dxxf

x

x




Adding adjacent areas




The Trapezoidal Rule

2

2

1212

0101

yyxx

yyxxI

Integrating from x0 to x2:

2

212112101001 yxxyxxyxxyxxI





hxxxx 1201

If the points are equidistant

2

2110 hyhyhyhyI

210 22

yyyh

I




Dividing the whole interval into “n”

subintervals

n

n

i

i yyyh

I1

1

0 22




The Algorithm

• To integrate f(x) from a to b, determine the number of intervals “n”

• Calculate the interval length h=(b-a)/n

• Evaluate the function at the points yi=f(xi) where xi=x0+i*h

• Evaluate the integral by performing the summation

n

n

i

i yyyh

I1

1

0 22




Note that

X0=a

Xn=b




Example

• Integrate

• Using the trapezoidal

rule

• Use 2,3,&4 points and

compare the results

1

0

2dxxI




Solution

• Using 2 points (n=1),

h=(1-0)/(1)=1

• Substituting:

212

1yyI 5.010

2

1I

Y X

0 0

1 1

2 points, 1 interval




Solution


h=(1-0)/(2)=0.5

• Substituting:

321 22

5.0yyyI

375.0125.0*202

5.0I

Y X

0 0

0.25 0.5

1 1





Solution


h=(1-0)/(3)=0.333

• Substituting:

4321 222

333.0yyyyI

3519.01444.0*2111.0*202

333.0I

Y X

0 0

0.111 0.33

0.444 0.667

1 1





Let’s use Interpolation!




Interpolation!

• If we have a function that needs to be integrated between two points

• We may use an approximate form of the function to integrate!

• Polynomials are always integrable

• Why don’t we use a polynomial to approximate the function, then evaluate the integral




Example

• To perform the

definite integration of

the function between

(x0 & x1), we may

interpolate the

function between the

two points as a line.

0

01

010 xx

xx

yyyxf




Example

• Performing the integration on the approximate

function:

1

0

1

0

0

01

010

x

x

x

x

dxxxxx

yyydxxfI

1

0

0

2

01

010

2

x

x

xxx

xx

yyxyI




Example

• Performing the integration on the approximate

function:

00

2

0

01

010010

2

1

01

0110

22xx

x

xx

yyxyxx

x

xx

yyxyI

2

0101

yyxxI

• Which is equivalent to the area of the trapezium!





2

0101

yyxxI

2

2

1212

0101

yyxx

yyxxI

Integrating from x0 to x2:




Simpson’s Rule

Using a parabola to join three

adjacent points!




Quadratic Interpolation

• If we get to interpolate a quadratic equation

between every neighboring 3 points, we may use

Newton’s interpolation formula:

103021 xxxxbxxbbxf

1010

2

3021 xxxxxxbxxbbxf




Integrating

1010

2

3021 xxxxxxbxxbbxf

2

0

2

0

1010

2

3021

x

x

x

x

dxxxxxxxbxxbbdxxf

2

0

2

0

10

2

10

3

30

2

21232

x

x

x

x

xxxx

xxx

bxxx

bxbdxxf




After substitutions and

manipulation!

210 43

2

0

yyyh

dxxf

x

x




Working with three points!

210 43

2

0

yyyh

dxxf

x

x




For 4-Intervals

432210 443

4

0

yyyyyyh

dxxf

x

x




In General: Simpson’s Rule

n

n

i

i

n

i

i

x

x

yyyyh

dxxfn 2

,..4,2

1

,..3,1

0 243

0

NOTE: the number of intervals HAS TO BE even




Example

• Integrate

• Using the Simpson

rule

• Use 3 points

1

0

2dxxI




Solution


h=(1-0)/(2)=0.5

• Substituting:

• Which is the exact

solution!

210 43

5.0yyyI

3

1125.0*40

3

5.0I

Education

Numerical Integration