This Presentation Is Specially Made For Those Engineering Students Who are In Gujarat Technological University. This Presentation Clears Your All Doubts About Basics Fundamentals of Numerical Integration. Also You Will Learn Different Types Of Error Formula To Solve the Numerical Integration Sum.

1. ACTIVE LEARNINGASSIGNMENT PRESENTATION
COMPLEX VARIABLES & NUMERICAL METHODS (2141906)
TOPIC
NUMERICAL INTEGRATION : ERROR FORMULAE,
GAUSSIAN QUADRATURE FORMULAE
PREPARED BY
MECHANICAL – 4B2
DEVANSU KHORASIYA (150120119066)
GUIDED BY
PROF. RAVI PANCHAL
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

2. INTRODUCTION
What is Numerical Integration?
 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.
 The basic problem considered by numerical
integration is to compute an approximate solution to a
definite integral. It is different from analytical integration in
two ways: first it is an approximation and will not yield an
exact answer; Error analysis is a very important aspect in
numerical integration. Second it does not produce an
elementary function with which to determine the area
given any arbitrary bounds; it only produces a numerical
value representing an approximation of area.
Let’s Start The Journey…
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

3. First Of All…
 History of Numerical
Integration 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.
 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 oftrapezoids 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.
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

4. PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

5.  Elements of Numerical Integration
If f(x) is a smooth well behaved Function, integrated over a small number of
dimensions and the limits of integration are bounded, there are many methods of
approximating the integral with arbitrary precision. We consider an indefinite integral:
Numerical integration methods can generally be described as combining evaluations of
the integrand to get an approximation to the integral.
In this example the definite integral is
thus approximated using areas of
rectangles. The integration points and
weights depend on the specific
method used and the accuracy
required from the approximation. An
important part of the analysis of any
numerical integration method is to
study the behavior of the
approximation error as a function of
the number of integrand evaluations.
𝑎
𝑏
𝑓 𝑥 𝑑𝑥
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

6. An Easy Method of Numerical Integration: Trapezoid Rule
The Trapezoid Rule calls for the approximation of area under a curve by fitting
trapezoids under the curve and regularly spaced intervals. This method is very common
in beginning calculus courses used as a transition into analytical integration. The
method uses the outputs of the function as the two legs of the trapezoid and the
specified interval is the height. The area of a trapezoid is one half the height multiplied
by the sum of the two bases:
It would be more advantageous to use
more trapezoids of smaller height to better
fit the curvature of the graph. As we
increase the number of trapezoids by
increasing the number of divisions in
the interval, accuracy increases.
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

7. SIMPSON’S
𝟏
𝟑
RULE
𝒙 𝟎
𝒙 𝟎+𝒏𝒉
𝒇(𝒙)dx =
𝟓
𝟗
[(𝒚 𝟎 + 𝒚 𝒏) + 4(𝒚 𝟏+ 𝒚 𝟑 + ⋯ + 𝒚 𝒏−𝟏 ) + 2((𝒚 𝟐+ 𝒚 𝟒 + ⋯ + 𝒚 𝒏−𝟐 ) ]
SIMPSON’S
𝟑
𝟖
RULE
𝒙 𝟎
𝒙 𝟎+𝒏𝒉
𝒇(𝒙)dx =
𝟑𝒉
𝟖
[(𝒚 𝟎 + 𝒚 𝒏) + 2(𝒚 𝟑+ 𝒚 𝟔 + ⋯ + 𝒚 𝒏−𝟑 ) + 3(𝒚 𝟏 + 𝒚 𝟐 + 𝒚 𝟒 … + 𝒚 𝒏−𝟏 ) ]
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

8.  Gaussian Quadrature Formulae
An n- Point Gaussian Quadrature Formula is a Quadrature Formula Constructed to give an
Exact Result For Polynomials degree 2n-1 or Less by a Suitable choice of the Points and
Weight 𝑤1for I = 1, 2, 3,….n.
The Gaussian Quadrature Formula is..
−1
1
𝑓(𝑥)dx = 𝑖=1
𝑛
𝑤𝑖f(𝑥𝑖)
One-Point Gaussian Quadrature Formulae
Consider a Function f(x) over the Interval [-1, 1] with Sampling Point 𝑥1. and
Weight 𝑤1 Respectively.
The One Point Gaussian Quadrature Formula is..
−1
1
𝑓(𝑥)dx = 𝑤1f(𝑥1)
For n = 1 We Find Polynomials 2n-1 = 2(1)-1 = 1 and For f(x) = 1 or x…
−𝟏
𝟏
𝒇(𝒙)dx = 2f(0)
𝑥1
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

9.  Two-Point Gaussian Quadrature Formulae
Consider a Function f(x) over the Interval [-1, 1] with Sampling Points 𝑥1, 𝑥2. and
Weights 𝑤1, 𝑤2 Respectively.
The Two Point Gaussian Quadrature Formula is..
−𝟏
𝟏
𝒇(𝒙)dx = 𝒘 𝟏f(𝒙 𝟏) + 𝒘 𝟐f(𝒙 𝟐)
This Formula Will be Exact for Polynomials of Degree up to 2n-1 = 2(2)-1 = 3, i.e.
It is Exact for f(x)= 1, x, 𝑥2
𝑎𝑛𝑑 𝑥3
.
−𝟏
𝟏
𝒇(𝒙)dx = f(-
𝟏
𝟑
) + f(
𝟏
𝟑
)
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

10.  Three-Point Gaussian Quadrature Formulae
Consider a Function f(x) over the Interval [-1, 1] with Sampling Points 𝑥1, 𝑥2, 𝑥3. and
Weights 𝑤1, 𝑤2, 𝑤3Respectively.
The Three Point Gaussian Quadrature Formula is..
−𝟏
𝟏
𝒇(𝒙)dx = 𝒘 𝟏f(𝒙 𝟏) + 𝒘 𝟐f(𝒙 𝟐) +𝒘 𝟑f(𝒙 𝟑)
This Formula Will be Exact for Polynomials of Degree up to 2n-1 = 2(3)-1 = 5,
i.e.
It is Exact for f(x)= 1, x, 𝒙 𝟐, 𝒙 𝟑, 𝒙 𝟒 𝒂𝒏𝒅 𝒙 𝟓.
−𝟏
𝟏
𝒇(𝒙)dx =
𝟓
𝟗
f(-
𝟑
𝟓
) +
𝟖
𝟗
f(0) +
𝟓
𝟗
f(
𝟑
𝟓
)
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

11. EXAMPLE :
−𝟏
𝟏 𝒅𝒙
𝟏+ 𝒙 𝟐 by One Point, Two Point and Three point Gaussian
Formulae.
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

12. PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

13. ANY QUESTIONS ???
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

14. ATTHE LAST…….
One Humble Request to all ofYOU….
Make sureYour House, School, College, Society, Road,
State, Nation, World is Neat and Clean……PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)

15. THANK YOU GUYS FOR LISTINING ME……
PRESENTATION & DESIGNED BY
DEVANSU KHORASIYA
(150120119066)
MYAIM IS….
CLEAN INDIA
GREEN INDIA &
MAKE IN INDIA.
PREPARED BY : DEVANSU KHORASIYA(GIT-150120119066)