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Numerical Integration
• In general, a numerical integration is the approximation
of a definite integration by a “weighted” sum of function
values at discretized points within the interval of
integration.

∫

b

a

N

f ( x )dx ≈ ∑ wi f ( xi )
i =0

where wi is the weighted factor depending on the integration
schemes used, and f ( xi ) is the function value evaluated at the
given point xi
Rectangular Rule

f(x)
height=f(x1*)

x=a
x=x1*

∫

b

a

height=f(xn*)

x=b

x

Approximate the integration,
b
∫a f ( x)dx , that is the area under
the curve by a series of
rectangles as shown.
The base of each of these
rectangles is ∆x=(b-a)/n and
its height can be expressed as
f(xi*) where xi* is the midpoint
of each rectangle

x=xn*

f ( x )dx = f ( x1*) ∆x + f ( x2 *) ∆x + .. f ( xn *) ∆x

= ∆x[ f ( x1*) + f ( x2 *) + .. f ( xn *)]
Trapezoidal Rule

f(x)

x=a
x=x1

x=b
x=xn-1

x

The rectangular rule can be made
more accurate by using
trapezoids to replace the
rectangles as shown. A linear
approximation of the function
locally sometimes work much
better than using the averaged
value like the rectangular rule
does.

∆x
∆x
∆x
∫a f ( x )dx = 2 [ f (a ) + f ( x1 )] + 2 [ f ( x1 ) + f ( x2 )] + .. + 2 [ f ( xn−1) + f (b)]
1
1
= ∆x[ f (a ) + f ( x1 ) + .. f ( xn −1 ) + f (b)]
2
2
b
Simpson’s Rule
Still, the more accurate integration formula can be achieved by
approximating the local curve by a higher order function, such as
a quadratic polynomial. This leads to the Simpson’s rule and the
formula is given as:
∆x
∫a f ( x )dx = 3 [ f (a ) + 4 f ( x1 ) + 2 f ( x2 ) + 4 f ( x3 ) + ..
..2 f ( x2 m−2 ) + 4 f ( x2 m −1 ) + f ( b)]
b

It is to be noted that the total number of subdivisions has to be an
even number in order for the Simpson’s formula to work
properly.
Examples
Integrate f ( x ) = x 3 between x = 1 and x = 2.

∫

2

1

1
1
2
x 3dx= x 4 |1 = (24 − 14 ) = 3.75
4
4

2-1
Using 4 subdivisions for the numerical integration: ∆x=
= 0.25
4
Rectangular rule:
i

xi*

f(xi*)

1

1.125 1.42

2

1.375 2.60

3

1.625 4.29

4

1.875 6.59

∫

2

1

x 3dx

= ∆x[ f (1.125) + f (1.375) + f (1.625) + f (1.875)]
= 0.25(14.9) = 3.725
Trapezoidal Rule
i

xi
1

f(xi)
1

1 1.25

1.95

2 1.5

3.38

3 1.75

∫

2

1

x 3dx

1
1
f (1) + f (1.25) + f (1.5) + f (1.75) + f (2)]
2
2
= 0.25(15.19) = 3.80
= ∆x[

5.36

2

8

Simpson’s Rule

∫

2

1

x 3dx =

∆x
[ f (1) + 4 f (1.25) + 2 f (1.5) + 4 f (1.75) + f (2)]
3

0.25
=
(45) = 3.75 ⇒ perfect estimation
3
Trapezoidal Rule
i

xi
1

f(xi)
1

1 1.25

1.95

2 1.5

3.38

3 1.75

∫

2

1

x 3dx

1
1
f (1) + f (1.25) + f (1.5) + f (1.75) + f (2)]
2
2
= 0.25(15.19) = 3.80
= ∆x[

5.36

2

8

Simpson’s Rule

∫

2

1

x 3dx =

∆x
[ f (1) + 4 f (1.25) + 2 f (1.5) + 4 f (1.75) + f (2)]
3

0.25
=
(45) = 3.75 ⇒ perfect estimation
3

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Numerical integration

  • 1. Numerical Integration • In general, a numerical integration is the approximation of a definite integration by a “weighted” sum of function values at discretized points within the interval of integration. ∫ b a N f ( x )dx ≈ ∑ wi f ( xi ) i =0 where wi is the weighted factor depending on the integration schemes used, and f ( xi ) is the function value evaluated at the given point xi
  • 2. Rectangular Rule f(x) height=f(x1*) x=a x=x1* ∫ b a height=f(xn*) x=b x Approximate the integration, b ∫a f ( x)dx , that is the area under the curve by a series of rectangles as shown. The base of each of these rectangles is ∆x=(b-a)/n and its height can be expressed as f(xi*) where xi* is the midpoint of each rectangle x=xn* f ( x )dx = f ( x1*) ∆x + f ( x2 *) ∆x + .. f ( xn *) ∆x = ∆x[ f ( x1*) + f ( x2 *) + .. f ( xn *)]
  • 3. Trapezoidal Rule f(x) x=a x=x1 x=b x=xn-1 x The rectangular rule can be made more accurate by using trapezoids to replace the rectangles as shown. A linear approximation of the function locally sometimes work much better than using the averaged value like the rectangular rule does. ∆x ∆x ∆x ∫a f ( x )dx = 2 [ f (a ) + f ( x1 )] + 2 [ f ( x1 ) + f ( x2 )] + .. + 2 [ f ( xn−1) + f (b)] 1 1 = ∆x[ f (a ) + f ( x1 ) + .. f ( xn −1 ) + f (b)] 2 2 b
  • 4. Simpson’s Rule Still, the more accurate integration formula can be achieved by approximating the local curve by a higher order function, such as a quadratic polynomial. This leads to the Simpson’s rule and the formula is given as: ∆x ∫a f ( x )dx = 3 [ f (a ) + 4 f ( x1 ) + 2 f ( x2 ) + 4 f ( x3 ) + .. ..2 f ( x2 m−2 ) + 4 f ( x2 m −1 ) + f ( b)] b It is to be noted that the total number of subdivisions has to be an even number in order for the Simpson’s formula to work properly.
  • 5. Examples Integrate f ( x ) = x 3 between x = 1 and x = 2. ∫ 2 1 1 1 2 x 3dx= x 4 |1 = (24 − 14 ) = 3.75 4 4 2-1 Using 4 subdivisions for the numerical integration: ∆x= = 0.25 4 Rectangular rule: i xi* f(xi*) 1 1.125 1.42 2 1.375 2.60 3 1.625 4.29 4 1.875 6.59 ∫ 2 1 x 3dx = ∆x[ f (1.125) + f (1.375) + f (1.625) + f (1.875)] = 0.25(14.9) = 3.725
  • 6. Trapezoidal Rule i xi 1 f(xi) 1 1 1.25 1.95 2 1.5 3.38 3 1.75 ∫ 2 1 x 3dx 1 1 f (1) + f (1.25) + f (1.5) + f (1.75) + f (2)] 2 2 = 0.25(15.19) = 3.80 = ∆x[ 5.36 2 8 Simpson’s Rule ∫ 2 1 x 3dx = ∆x [ f (1) + 4 f (1.25) + 2 f (1.5) + 4 f (1.75) + f (2)] 3 0.25 = (45) = 3.75 ⇒ perfect estimation 3
  • 7. Trapezoidal Rule i xi 1 f(xi) 1 1 1.25 1.95 2 1.5 3.38 3 1.75 ∫ 2 1 x 3dx 1 1 f (1) + f (1.25) + f (1.5) + f (1.75) + f (2)] 2 2 = 0.25(15.19) = 3.80 = ∆x[ 5.36 2 8 Simpson’s Rule ∫ 2 1 x 3dx = ∆x [ f (1) + 4 f (1.25) + 2 f (1.5) + 4 f (1.75) + f (2)] 3 0.25 = (45) = 3.75 ⇒ perfect estimation 3