This document discusses numerical integration techniques, including the trapezoidal rule and Simpson's rule. It begins by establishing the need for numerical integration when exact integrals cannot be calculated. It then derives the trapezoidal rule using geometric insight by approximating the area under a curve as trapezoids. The document explains how to apply the trapezoidal rule using equidistant points and presents an example. Finally, it introduces Simpson's rule, which uses quadratic interpolation between three points to better approximate the area under a curve compared to the trapezoidal rule. Students are assigned related homework problems.

1. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Numerical Integration

2. ENEM602 Spring 2007
Dr. Eng. 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

3. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Need for Numerical Integration!
( )
6
11
01
2
1
3
1
23
1
1
0
231
0
2
=−





++=






++=++= ∫ x
xx
dxxxI
( ) 11
0
1
0
1 −−−
−=−== ∫ eedxeI xx
∫
−
=
1
0
2
dxeI x

4. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
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?

5. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
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:
( ) ( )01
01
2
1
0
xx
yy
dxxf
x
x
−
+
≈∫

6. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Adding adjacent areas

7. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
The Trapezoidal Rule
( ) ( )
( ) ( )
2
2
12
12
01
01
yy
xx
yy
xxI
+
−+
+
−≈
Integrating from x0 to x2:
( ) ( ) ( ) ( )
2
212112101001 yxxyxxyxxyxx
I
−+−+−+−
≈

8. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
The Trapezoidal Rule
( ) ( ) hxxxx =−=− 1201
If the points are equidistant
2
2110 hyhyhyhy
I
+++
≈
( )210 2
2
yyy
h
I ++≈

9. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Dividing the whole interval into “n”
subintervals






++≈ ∑
−
=
n
n
i
i yyy
h
I
1
1
0 2
2

10. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
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 yyy
h
I
1
1
0 2
2

11. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Note that
X0=a
Xn=b

12. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
• Integrate
• Using the trapezoidal
rule
• Use 2,3,&4 points and
compare the results
∫=
1
0
2
dxxI

13. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Solution
• Using 2 points (n=1),
h=(1-0)/(1)=1
• Substituting:
( )21
2
1
yyI +≈ ( ) 5.010
2
1
=+≈I
X Y
0 0
1 1
2 points, 1 interval

14. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Solution
• Using 3 points (n=2),
h=(1-0)/(2)=0.5
• Substituting:
( )321 2
2
5.0
yyyI ++≈
( ) 375.0125.0*20
2
5.0
=++≈I
X Y
0 0
0.5 0.25
1 1
3 points, 2 interval

15. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Solution
• Using 4 points (n=3),
h=(1-0)/(3)=0.333
• Substituting:
( )4321 22
2
333.0
yyyyI +++≈
( ) 3519.01444.0*2111.0*20
2
333.0
=+++≈I
X Y
0 0
0.33 0.111
0.667 0.444
1 1
4 points, 3 interval

16. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Let’s use Interpolation!

17. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
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

18. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
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
01
0 xx
xx
yy
yxf −
−
−
+≈

19. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
• Performing the integration on the approximate
function:
( ) ( )∫∫ 





−
−
−
+≈=
1
0
1
0
0
01
01
0
x
x
x
x
dxxx
xx
yy
ydxxfI
1
0
0
2
01
01
0
2
x
x
xx
x
xx
yy
xyI 













−
−
−
+≈

20. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
• Performing the integration on the approximate
function:
















−
−
−
+−
















−
−
−
+≈ 00
2
0
01
01
0010
2
1
01
01
10
22
xx
x
xx
yy
xyxx
x
xx
yy
xyI
( ) ( )
2
01
01
yy
xxI
+
−≈
• Which is equivalent to the area of the trapezium!

21. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
The Trapezoidal Rule
( ) ( )
2
01
01
yy
xxI
+
−≈
( ) ( )
( ) ( )
2
2
12
12
01
01
yy
xx
yy
xxI
+
−+
+
−≈
Integrating from x0 to x2:

22. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Simpson’s Rule
Using a parabola to join three
adjacent points!

23. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
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 ++−+−+≈

24. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
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
21
232
x
x
x
x
xxx
x
xx
x
bxx
x
bxbdxxf 











++−+





−+≈∫

25. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
After substitutions and
manipulation!
( ) [ ]210 4
3
2
0
yyy
h
dxxf
x
x
++≈∫

26. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Working with three points!
( ) [ ]210 4
3
2
0
yyy
h
dxxf
x
x
++≈∫

27. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
For 4-Intervals
( ) [ ]432210 44
3
4
0
yyyyyy
h
dxxf
x
x
+++++≈∫

28. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
In General: Simpson’s Rule
( ) 





+++≈ ∑∑∫
−
=
−
=
n
n
i
i
n
i
i
x
x
yyyy
h
dxxf
n 2
,..4,2
1
,..3,1
0 24
30
NOTE: the number of intervals HAS TO BE even

29. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
• Integrate
• Using the Simpson
rule
• Use 3 points
∫=
1
0
2
dxxI

30. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Solution
• Using 3 points (n=2),
h=(1-0)/(2)=0.5
• Substituting:
• Which is the exact
solution!
( )210 4
3
5.0
yyyI ++≈
( )
3
1
125.0*40
3
5.0
=++≈I

31. ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Homework #7
• Chapter 21, p. 610, numbers:
21.5, 21.6, 21.10, 21.11.