1. The document discusses numerical integration techniques for approximating definite integrals that cannot be solved analytically. It covers basic techniques like the rectangle, midpoint, and trapezoid rules as well as more accurate techniques like Simpson's rule. 2. Examples are provided to demonstrate calculating definite integrals numerically to approximate values like the natural logarithm of numbers. The document also introduces Monte Carlo integration techniques using random sampling. 3. As an example problem, the document calculates the final speed of a box moving under a time-varying force using numerical integration over the integral expression for work. The Simpson's rule is identified as an approach to implement in a programming code to solve this example.

1. 1
1
Programming, Computation, Simulation
Applications in Math & Physics
Numerical Integrations
Syeilendra Pramuditya
Department of Physics
Institut Teknologi Bandung

2. 2
Physics Problem
 Period of Pendulum
How Easy!
2
2
( )
( )
2
1
2
d t
t SHM
dt
g
f
l
l
T
f g

 
 

  
 
 

3. 3
Physics Problem
 Large Amplitude Pendulum..?

4. 4
Physics Problem
 Large Amplitude Pendulum..?
sin
g
d d
l
    
 
 
 

5. 5
Large Amplitude Pendulum
Problem: Division by zero

6. 6
Large Amplitude Pendulum
Legendre Complete Elliptic Integral of the First Kind
Trigonometric identities, change of variable, etc…

7. 7
Large Amplitude Pendulum
 Input
 g, l, 0
 Process
 Evaluate the integral
 Output
 Period of the pendulum
Small Amplitude
1
2
l
T
f g

 
Solution by series expansion
Check your result

8. 8
Do The Code
 Benchmark your integrator code first
0
1 0
sin(30 ) 0.5
sin (0.5) 30




9. 9
9
Numerical Integration
 Definite integral of f(t) on
interval [a,b]
 Analytic method: anti-
derivative
 F where F’(t) = f(t)
 Iab = F(b) – F(a)
( ) ( ) ( ) ( )
b
ab
a
F t I f t dt F b F a
   


10. 10
Hand Calculation?
 Calculate until 2 decimal numbers?
 a = 88/17
 b = square-root of 23
 c = ln(7)

11. 11
11
Numerical Integration
 ln7 = .. ?
 ln5 = .. ?
7
5
1
( )
1
ln 7 ln5
ab
f t
t
I dt
t

  

 Need to somehow calculate the numerical value of natural
logarithm
 Real-world functions don’t have anti derivatives
 Direct calculation of definite integrals is needed

12. 12
Numerical Integration
 Newton-Cotes Formula
 Composite Method
 Quadrature Method
 Riemann Sum
 Etc…

13. 13
13
Numerical Integration
 Example: value of ln(1.2)
1
1
ln( )
x
x dt
t
 
1.2
1
1
ln(1.2) dt
t
 
 Ln(1.2) = area below the curve

14. 14
14
Numerical Integration
 General Formula
0
( ) ( )
b n
ab i i
i
a
I f t dt f t w

  

 f(ti) : function f evaluated at ti
 wi : weighting factor

15. 15
Numerical Integration
 Calculating Area

16. 16
16
Rectangle Rule
( ) ( ) ( ) ( )
b
ab i i
a
I f t dt f t w f a b a
    


17. 17
17
Midpoint Rule
( ) ( ) ( )
2
b
ab i i
a
a b
I f t dt f t w f b a

 
    
 
 


18. 18
18
Trapezoid Rule
   
( ) ( ) ( )
2
b
ab i i
a
f a f b
I f t dt f t w b a

    


19. 19
19
Non-Linear Rule (Simpson)
Approx. by 3-point Lagrange interpolation (Quadratic)
2
( )
p x ax bx c
  

20. 20
20
Simpson Rule
( ) ( ) ( ) 4 ( )
6 2
b
ab i i
a
b a a b
I f t dt f t w f a f f b
   
 
     
 
 
 
 

Derived using integ. by substitution for 3-point Lagrange interpolation (Quadratic)

21. 21
21
Simpson Rule
1
1
1
( ) ( ) 4 ( )
6 2
b N
i i
ab i i
i
a
x x
x
I f x dx f x f f x



 
 


    
 
 
 
 


Derived using integ. by substitution for 3-point Lagrange interpolation (Quadratic)
N partitions

22. 22
More Complex Integral
7.6
3.2
2
( )
ln( )
( )
5
ab
I f t dt
x
f t
x





23. 23
Adaptive Method
 Automatic Re-partitioning

24. 24
Monte Carlo Integration
1
Average of 2, 8, 5, 12, 4
2 8 5 12 4
Average
5
1
( ) ( )
( )
1
( ) ( )
N
b
i
a
i
b
b
b a
b
a
a
a
f x f x
N
f x dx
f x f x dx
b a
dx

   


 





Discrete Functions
Continuous Functions

25. 25
Monte Carlo Integration
( )
1
( ) ( )
b
b
b a
b
a
a
a
f x dx
f x f x dx
b a
dx
 




1
1
( ) ( )
N
b
i
a
i
f x f x
N 
 
1
( ) ( ) ( ) ( )
b N
b
i
a
i
a
b a
f x dx b a f x f x
N 

    

Rectangle of same area with original function

26. 26
Do The Code: Monte Carlo Integration
5.7
1
1
ln(5.7) 1.74
dt
t
 


27. 27
Do The Code: Monte Carlo Integration
N ln(5.7)
10 1.669
20 1.914
40 1.622
80 1.872
160 1.819
320 1.690
640 1.693
1280 1.723
2560 1.739 1.600
1.650
1.700
1.750
1.800
1.850
1.900
1.950
1 10 100 1000 10000
N points
Numerical
Value
5.7
1
1
ln(5.7) 1.74
dt
t
 


28. 28
Do The Code: Monte Carlo Integration
5.7
1
1
ln(5.7) 1.74
dt
t
 

N ln(5.7) ln(5.7) ln(5.7)
10 1.669 2.087 1.651
20 1.914 1.936 1.670
40 1.622 2.043 1.901
80 1.872 1.787 1.759
160 1.819 1.675 1.789
320 1.690 1.777 1.653
640 1.693 1.726 1.705
1280 1.723 1.751 1.712
2560 1.739 1.738 1.740
1.500
1.600
1.700
1.800
1.900
2.000
2.100
2.200
1 10 100 1000 10000
N points
Numerical
value

29. 29
Do The Code: Monte Carlo Integration
5.7
1
1
ln(5.7) 1.74
dt
t
 

1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
0 20 40 60 80 100
N=10
N=160
N=2560

30. 30
Do The Code: Monte Carlo Integration
7.6
3.2
2
( )
ln( )
( )
5
ab
I f t dt
x
f t
x





31. 31
Sample Problem
A variable force is applied to an initially stationary box of mass
2.35 kg at x= 0 m. The box moves in +x direction due to the
applied force. What is the speed of the box when its position is at x
= 7.53 m? The surface is frictionless.
2 ˆ
( ) 5 sin ( rad)
F x x x i N
 

2 2
1 1
2 2
2 2
1 1
( ) ( )cos(0)
1
( ) 5 sin ( )
2
f i
x x
f
x x
x x
f
x x
K K K W
K F x dx F x dx
mv F x dx x x dx
   
 
  
 
 



32. 32
The Integral
2
2
1
5 sin ( )
x
x
I x x dx
 

Gnuplot
set xrange [0:10]
set xrange [0:20]
plot x*(sin(x))**2

33. 33
Practice
 Develop your integrator code using Simpson rule in Pascal
language
 The code mainly evaluates numerically:
 What is the final speed of the box?
 Using 75 partitions
2
2
1
5 sin ( )
x
x
I x x dx
 
