The Islamic University of Gaza Faculty of Engineering Civil Engineering Department

The Islamic University of GazaFaculty of Engineering

Civil Engineering Department

Numerical Analysis ECIV 3306

Chapter 22

Integration of Equations

Gauss Quadrature

• Gauss quadrature implements a strategy of positioning any two points on a curve to define a straight line that would balance the positive and negative errors.

• Hence, the area evaluated under this straight line provides an improved estimate of the integral.

Two points Gauss-Legendre Formula• Assume that the two Integration points are xo and x1 such that:

• The object of Gauss quadrature is to determine the equations of the form:

• c0 and c1 are constants, the function arguments x0 and x1 are unknowns…….(4 unknowns)

)()( 1100 xfcxfcI

Two points Gauss-Legendre Formula

• Thus, four unknowns to be evaluated require four conditions.

• If this integration is exact for a constant, 1st order, 2nd order, and 3rd order functions:

1

1

31100

1

1

21100

1

11100

1

11100

0)()(

32)()(

0)()(

21)()(

dxxxfcxfc

dxxxfcxfc

dxxxfcxfc

dxxfcxfc

Two points Gauss-Legendre Formula

• Solving these 4 equations, we can determine c1, c2, x1 and x2.

31

31 ffI

5773503.03

1

5773503.03

1:are pointsn Integratio The

1:are factors weightingThe

1

0

10

x

x

cc

Two points Gauss-Legendre Formula• Since we used limits for the previous integration from –1 to 1

and the actual limits are usually from a to b, then we need first to transform both the function and the integration from the x-system to the xd-system

1ax1bx

d2

abdx

2ab

2abx

f(x)

x

a b

f(xo)

f(x1)

xo x1 -1 1

f()

Higher-Points Gauss-Legendre Formula

)(.....)()()(

)(

nn2211

n

1iii

1

1i

fcfcfcfcI

:points Gauss n usingdfI

Multiple Points Gauss-LegendrePoints Weighting factor Function argument Exact

for 2 1.0 -0.577350269 up to

3rd 1.0 0.577350269 degree

3 0.5555556 -0.774596669 up to 5th

0.8888889 0.0 degree0.5555556 0.774596669

4 0.3478548 -0.861136312 up to 7th

0.6521452 -0.339981044 degree0.6521452 0.3399810440.3478548 0.861136312

6 0.1713245 -0.932469514 up to 11th

0.3607616 -0.661209386 degree0.4679139 -0.2386191860.4679139 0.2386191860.3607616 0.6612093860.1713245 0.932469514

Gauss Quadrature - Example

Find the integral of: f(x) = 0.2 + 25 x – 200 x2 + 675 x3 – 900 x4 + 400 x5

Between the limits 0 to 0.8 using:

– 2 points integration points (ans. 1.822578)

– 3 points integration points (ans. 1.640533)

Improper Integral• Improper integrals can be evaluated by making a change

of variable that transforms the infinite range to one that is finite,

b

A

b A

b

a

a

b

dxxfdxxfdxxf

abdtt

ft

dxxf

)()()(

011)(/1

/12

A

A

dtt

ft

dxxf0

/12

11)(Can be evaluated by Newton-Cotes closed formula

Improper Integral - Examples

• .

• .

• .

dtt

dtt

ttxx

dx

5.0

0

5.0

0 2

2 211

2/11)(1

)2(

dyyedyyedyye yyy

2

22

0

2

0

2 sin sin sin

dttet

dyye ty 2/1

0

2/12

2

2 )/1(sin1 sin

dyyedyyedyye yyy

2

2

2

2

dtet

dyye ty 2/1

0

/13

2 1

Multiple Integration

dydxyxfId

cy

b

ax

),(

• Double integral:

Multiple Integration using Gauss Quadrature Technique

• .

• .

• . ddf

2ab

2cddydxyxfI

1

1

1

1

d

cy

b

ax

),(),(

1ax1bx &

d2

abdx2

ab2

abx

&

1cx1dy &

d2

cddy2

cd2

cdy

&

Multiple Integration using Gauss Quadrature Technique

Now we can use the Gauss Quadrature technique:

If we use two points Gauss Formula:

n

1jjiij

n

1i

fcc2

ab2

cdI ),(

)}],(),({

)},(),({[

31

31fc

31

31fcc

31

31fc

31

31fcc

2ab

2cdI

212

211

2

1jjijj

2

1i

fcc2

ab2

cdI ),(

2

1jj2j1j 3

1fc3

1fcc2

ab2

cdI )],(),([

Double integral - Example

72222),( 22 yxxxyyxT

• Compute the average temperature of a rectangular heated plate which is 8m long in the x direction and 6 m wide in the y direction. The temperature is given as:

• (Use 2 segment applications of the trapezoidal rule in each dimension)

Double integral - Example

6667.58)86/(2816,28163/1

56)86/(2688,2688)2(

)72222( 6

0

8

0

22

avg

avg

TIruleSimpson

TInrulelTrapezoidaMultiple

dxdyyxxxyI

HW: Use two points Gauss formula to solve the problem