09 numerical integration

ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik

Numerical Integration


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

dxxfx

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

22110 hyhyhyhy

I

210 22

yyyh

I


Dividing the whole interval into “n” subintervals

n

n

ii yyy

hI

1

10 2

2


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

ii yyy

hI

1

10 2

2


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

XY

00

11

2 points, 1 interval


Solution

• Using 3 points (n=2), h=(1-0)/(2)=0.5

• Substituting:

321 22

5.0yyyI

375.0125.0*202

5.0I

XY

00

0.50.25

11



Solution

• Using 4 points (n=3), h=(1-0)/(3)=0.333

• Substituting:

4321 222

333.0yyyyI

3519.01444.0*2111.0*202

333.0I

XY

00

0.330.111

0.6670.444

11



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.

001

010 xx

xx

yyyxf


Example

• Performing the integration on the approximate function:

1

0

1

0

001

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

20

01

010010

21

01

0110 22

xxx

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

10102

3021 xxxxxxbxxbbxf


Integrating

10102

3021 xxxxxxbxxbbxf

2

0

2

0

10102

3021

x

x

x

x

dxxxxxxxbxxbbdxxf

2

0

2

0

10

2

10

3

30

2

21 232

x

x

x

x

xxxx

xxx

bxxx

bxbdxxf


After substitutions and manipulation!

210 43

2

0

yyyh

dxxfx

x


Working with three points!

210 43

2

0

yyyh

dxxfx

x


For 4-Intervals

432210 443

4

0

yyyyyyh

dxxfx

x


In General: Simpson’s Rule

n

n

ii

n

ii

x

x

yyyyh

dxxfn 2

,..4,2

1

,..3,10 24

30

NOTE: the number of intervals HAS TO BE even


Example

• Integrate• Using the Simpson

rule• Use 3 points

1

0

2dxxI


Solution

• Using 3 points (n=2), h=(1-0)/(2)=0.5

• Substituting:

• Which is the exact solution!

210 43

5.0yyyI

3

1125.0*40

3

5.0I


Homework #7

• Chapter 21, p. 610, numbers:21.5, 21.6, 21.10, 21.11.

09 numerical integration

Technology

points n

adjacent points

interval y x

points y i

trapezoidal rule use

xy plane

approximate function

simpson rule use