Numerical Integration
Dr. Asaf Varol
1
Numerical Integration
•
Numerical integration is a primary tool used for definite integrals that
cannot be solved analytically. A numerical integration rule has the
form
• we investigate several basic quadrature formulas that use function
values at equally spaced points; these methods are known as
Newton-Cotes formulas. There are two types of Newton –Cotes
formulas, depending on whether or not the function values at the
ends of the interval of integration are used. The trapezoid and
Simpson rules are examples of “closed” formulas, in which the
endpoint values are used. The midpoint rule is the simplest example
of an “open” formula, in which the endpoints are not used [2].
2
Newton-Cotes Closed Formulas
Trapezoid Rule
• One of the simplest ways to approximate the
area under a curve is to approximate the curve
by a straight line. The trapezoid rule
approximates the curve by the straight line that
passes through the points [a, f(a) and b, f(b)], the
two ends of the interval of interest. We have
x0=a, x1=b, and h=b-a, and then
3
Example
4
Matlab Program
5
Diagram
6
Newton-Cotes Closed Formulas
Simpson’s Rule
7
Newton-Cotes Closed Formulas
Simpson’s Rule (Cont’d)
8
1
Matlab Program
1
1 x
2
dx
0
9
1
Diagram
1
1 x
2
dx
0
10
Newton-Cotes Closed Formulas
Midpoint Rule
11
Figure

sin( x)
dx
x
0
S
• Figure given on the
right side compares
the actual value of the
area with that found
by using the midpoint
rule. The area given
by the integral S
(hatched) and the
approximation using
the midpoint rule
(shaded) [2].
12

Matlab Program
sin( x)
dx
x
0
S
13

Diagram
sin( x)
dx
x
0
S
14
Gaussian quadrature
15
Gaussian quadrature
16
Example
e x
2
17
End of Chapter 5
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References
•
•
•
•
•
Celik, Ismail, B., “Introductory Numerical Methods for Engineering Applications”,
Ararat Books & Publishing, LCC., Morgantown, 2001
Fausett, Laurene, V. “Numerical Methods, Algorithms and Applications”, Prentice Hall,
2003 by Pearson Education, Inc., Upper Saddle River, NJ 07458
Rao, Singiresu, S., “Applied Numerical Methods for Engineers and Scientists, 2002
Prentice Hall, Upper Saddle River, NJ 07458
Mathews, John, H.; Fink, Kurtis, D., “Numerical Methods Using MATLAB” Fourth
Edition, 2004 Prentice Hall, Upper Saddle River, NJ 07458
Varol, A., “Sayisal Analiz (Numerical Analysis), in Turkish, Course notes, Firat
University, 2001
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