Numerical Integration Project Report

COMPUTING
METHODS
K16-3791 Uzair Hussain K16-3778
Bilal Khan K16-3769
Muzammil Asrar K15-2251 Sanjay
Babu
SIR JAMIL USMANI
------------------------
FAST NUCES
10/5/2018

CONTENTS:
 History and background of Numerical
Integration
 Flowchart, Algorithms and Programs
 Pros & Cons of Numerical Integration
 Summary
 References

HISTORY OF
NUMERICAL
INTEGRATION
The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation
and Numeric Integration for the Mathematical Laboratory by David Gibb.
The beginnings of numerical integration have its roots in antiquity. A prime example of
how ancient these methods are, is the Greek quadrature of the circle by means of inscribed and
circumscribed regular polygons. This process led Archimedes to an upper bound and lower bound for the value Pi.
These methods were used widely due to the lack of formal calculus. The method
of the sum of an infinitesimal area over a finite range was unknown until the sixteenth century
when Newton formalized the concepts of what we know now know as calculus. The earliest forms
of numerical integration are similar to that of the Greek method of inscribing regular polygons
into curved functions. This process broken down was taking a known area and overlapping it with
an unknown area to approximate the area of the unknown shape. One could improve accuracy by
choosing a better fitting shape. Later methods decided to improve upon estimating area under a
curve decided to use more polygons but smaller in area. Such an example is the use of rectangles
evenly spaced under a curve to estimate the area. Even further improvements saw the use of
trapezoids instead of rectangles to better fit the curvature of the function being analyzed. Today
the best methods for numerical integration are known as quadrature methods that have a very
small error.

Start
Input ‘eq’
equation, limits
‘x0, x1’ & ‘N’
If N == 1
Formulate
Trapezoid
Equation
Formulate
Simpson’s
1/3 Equation
Formulate
Simpson’s
3/8 Equation
If N == 2
If N == 3
A
N
Yes
Yes
Yes
No
No
No
CLOSED
NEWTON
CROOTES

A
N
=
=
1
If N == 4
Formulate
Boole’s
Equation
Default
Invalid
Input
Show ‘ans’
Answer
End
End
Yes
Yes
No

Start
Input ‘eq’
equation, limits
‘x0, x1’ & ‘N’
If N == 1
Formulate
using
Formula 1
Formulate
using
Formula 2
Formulate
using
Formula 3
If N == 2
If N == 3
A
N
=
Yes
Yes
Yes
No
No
No
OPEN
NEWTON
CROOTES

A
N
=
=
1
If N == 4
Formulate
using
Formula 4
Default
Invalid
Input
Show ‘ans’
Answer
End
End
Yes
Yes
No

Start
Eq = equation input
N = respective formulae
X0 = input lower limit
X1 = input upper limit
H = (x1 – x0)/2
Switch case ‘N’
(N == 1)
answer = [h/2 * (res(x0) + res(x0+h))]
(N == 2)
answer = [h/3 * (res(x0) + 4*res(x0+h) + res(x0+2*h))]
(N == 3)
answer = [3*h/8 * (res(x0) + 3*res(x0+h) +
3*res(x0+2*h) + res(x0+3*h))]
(N == 4) answer = [ 2*h/45 * (7*res(x0) + 32*res(x0+h)
+ 12*res(x0+2*h) + 32*res(x0+3*h) + 7*res(x0+4*h))]
End switch case
Print answer
End.
CLOSED
NEWTON
CROOTES

]
Start
Eq = equation input
N = respective formulae
X0 = input lower limit
X1 = input upper limit
H = (x1 – x0)/2
Switch case ‘N’
(N == 1)
answer = [(2*h) * (res(x0))]
(N == 2)
answer = [(3*h)/2 * (res(x0) + res(x0+h))]
(N == 3)
answer = [(4*h)/3 * (2*res(x0) - res(x0+h) +
2*res(x0+2*h))]
(N == 4) answer = [5*h/24 * (11*res(x0) + res(x0+h) +
res(x0+2*h) + 11*res(x0+3*h))]
End switch case
Print answer
End.
OPEN
NEWTON
CROOTES

Reasons for numerical integration
There are several reasons for carrying out numerical integration.
1. The integrand f(x) may be known only at certain points, such as obtained by sampling. Some
embedded system and other computer applications may need numerical integration for this reason.
2. A formula for the integrand may be known, but it may be difficult or impossible to find an
antiderivative that is an elementary function. An example of such an integrand is f(x) = exp (−x2),
the antiderivative of which (the error function, times a constant) cannot be written in elementary
form.
3. It may be possible to find an antiderivative symbolically, but it may be easier to compute a
numerical approximation than to compute the antiderivative. That may be the case if the
antiderivative is given as an infinite series or product, or if its evaluation requires a special function
that is not available.
Applications:
it helps to
 Find the area
 Locate the centroid
 Find the arc length of graph
 Find the surface area of solid
 Find the volume of a solid figure
 Solve for the work done
 Solve the moment of inertia
It is also used to find
 Water plane area
 Sectional area
 Submerged volume
 Longitudinal center of floatation
Pros
PROS & CONS
FOR NUMERICAL
INTEGRATION

 Possible to integrate any function
 Multidimensional integrals are straightforward
 Ability to solve integrals along irregular domains in multidimensional spaces (any shape).
 Numerical integration gives you an answer to some problems that analytic techniques don’t.
Cons
 There is an intrinsic error in calculation
 Numerical integrals are, always, computationally expensive.

SUMMARY AND
REFRENCES
 Wikipedia
 Quora
 California University papers
 MATLAB Documentation
In numerical analysis, numerical integration constitutes a broad family of
algorithms for calculating the numerical value of a definite integral, and by
extension, the term is also sometimes used to describe the numerical solution of
differential equations. This article focuses on calculation of definite integrals. The term
numerical quadrature (often abbreviated to quadrature) is more or less a synonym
for numerical integration, especially as applied to one-dimensional integrals.
Integration has been there since even before the proper use of calculus. In modern
day integration has led to some great creations including the petronas towers and
the Sydney opera house.

Numerical Integration Project Report

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