The document discusses various numerical integration techniques for approximating definite integrals, including rectangular, trapezoidal, and Simpson's rules. It provides formulas for each technique and examples of their application. The techniques can be compared based on their accuracy and the value of 'n' used, with larger n values typically providing more accurate solutions. Sources are also cited for further information on numerical integration methods.
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Error analysis in numerical integration
1. FAZAIA COLLEGE OF EDUCATION FOR WOMEN
ASSIGNMENT OF NUMERICAL ANALYSIS (Advance Math
VIII)
TOPIC: NUMERICAL INTEGRATION
Submitted to:
Ma’am Mehak
Submitted by:
Amenah Gondal (EDU(S)-2017-F-11)
Class:
B.S.Ed Hons (VIII)
2. ABSTRACT
Numerical Integration is the approximation computation of an integral using numerical
techniques. The numerical computation of an integral is sometimes called quadrature. We can
use numerical integration ti estimate the values of definite integrals when a closed form of the
integral is difficult to find or when an approximate value only of integral is needed.
The most commonly rules for numerical integration are rectangular, trapezoidal, simpson1/3,
simpson 3/8, boole and weddle rules. We get formulas for these rules by using Newton Cotes
Quadrature Formula by putting values for ‘n’ i.e., n=1,2,…,6. and neglecting higher derivatives
according to the requirement. All these rules can be compared with each other and we can find
that the rule with greater value of ‘n’ has more accurate and exact solution.
NUMERICAL INTEGRATION
1- RECTANGULAR RULE
The formula for rectangular rule is
It is also called Mid-Point formula.
This rule approximates the area under the curve by rectangles whose height is the mid-point of
each sub-interval.
3. EXAMPLE:
Evaluate the integral using rectangular rule for n=16 and compare with exact
value.
SOLUTION:
Given a=0, b=1, n=16
x 0 0.0625 0.125 0.1875 0.25 0.3125 0.375 0.4375 0.5 0.5625 0.625 0.6875
F(x) 1 0.9961 0.9846 0.9660 0.9412 0.9110 0.8767 0.8393 0.8 0.7596 0.7596 0.6790
0.75 0.8125 0.875 0.9375 1
0.64 0.6024 0.5664 0.5322 0.5
By putting values, we get
4. 2-TRAPEZOIDAL RULE:
The formula for trapezoidal rule is
In this rule, we find area under a curve is evaluated by dividing the total area into little
trapezoids rather than rectangles. This rule is used for n=1 and its multiples.
EXAMPLE:
Calculate the integral where h=0.05 using trapezoidal rule and compare with
exact value.
SOLUTION:
Given a=1.0, b=1.30, h=0.05, n=7
x 1 1.05 1.1 1.15 1.2 1.25 1.30
F(x) 1 1.0246 1.0488 1.072 1.095 1.118 1.14
5. By putting values, we get
Exact Value:
3-SIMPSON’S RULE:
The formula for simpson’s rule is
Simpson’s 1/3 rule is an extension of the trapezoidal rule in which the integrand is approximated
by a second order polynomial. This rule approximates the definite integral by first approximating
the original function using piecewise quadratic functions. This rule is used for n=2 and its
multiples.
6. EXAMPLE:
Use simpson’s 1/3 rule for n=4 to calculate correct to four decimal places.
SOLUTION:
Given a=0, b=1, n=4
x 0 0.25 0.5 0.75 1
F(x) 0.7071 0.7012 0.6822 0.6500 0.6065
By putting values, we get
4-SIMPSON’S RULE:
The formula for simpson’s rule is
This rule relies on approximating the curve with a cubic polynomial. This rule is used for n=3
and its multiples.
7. EXAMPLE:
Evaluate using seven points simpson’s rule.
SOLUTION:
Given a=0.1, b=0.7, n=6
x 0.1 0.2 0.3 0.4 0.5 0.6 0.7
F(x) 0.0997 0.1974 0.2915 0.3805 0.4636 0.5404 0.6107
By putting values, we get
5-BOOLE’S RULE:
The formula for boole’s rule is
This rule can be approximated by a polynomial of 4th
degree so that 5th
and higher derivatives are
vanishes. It can be used for the subinterval i.e., n=4. This rule is used for n=4 and its multiples.
8. EXAMPLE:
Apply nine points Boole’s rule to evaluate .
SOLUTION:
Given a=0.3, b=0.7, n=8
x 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
F(x) 2.5377 2.5576 2.5791 2.6021 2.6265 2.6525 2.6799 2.7089 2.7893
By putting values, we get
6-WEDDLE’S RULE:
The formula for weddle’s rule is
This rule approximating the integral of f(x) by giving n=6. It is used to solve multiple integrals.
It can only be used for the sub interval i.e., n=6.
9. EXAMPLE:
Apply seven point weddles’s rule to evaluate
SOLUTION:
Given a=0, b= , n=6,
x 0
F(x) 0 3.2817 0.42303 0.38898 0.27987 0.13820 0
By putting values, we get
TWO-POINT GUASSIAN QUADRATURE FORMULA
The formula for two point guassian quadrature formula
We use two-point guassian quadrature formula for two point interval i.e., -1 to 1. We use this
formula for evaluating integral without solving integration.
EXAMPLE:
Evaluate using two points guassian quadrature formula.
SOLUTION:
Given
a=0, b=1
10.
11. ERROR ANALYSIS IN NUMERICAL INTEGRATION
Let be the integrand i.e., the function to be integrated within the limits say either
or where such that
stands for error in the exact value and the approximate value i.e.,
ERROR TERM IN RECTANGULAR RULE:
The amount of error in rectangular rule is
12. ERROR TERM IN TRAPEZOIDAL RULE:
The magnitude of error in trapezoidal rule is
16. FORMULAS
1. Rectangular Rule:
2. Trapezoidal Rule:
3. Simpson’s 1/3 Rule:
4. Simpson’s 3/8th Rule:
5. Boole’s Rule:
6. Weddle’s Rule:
7. Two Point Guassian Quadrature Formula
17. REFERENCES
1. Iqbal, D. (n.d.). An introduction to Numerical Analysis. Lahore: Ilmi Kitab Khana.
2. Kiran, E. (2015, 5 22). Slideshare. Retrieved from https://www.slideshare.net
3. N.Shah, P. (2021). AtoZ Maths. Retrieved from https://atozmath.com
4. Saleem, M. (n.d.). Numerical Analysis II. Muzammil Tanveer.
5. Wikipedia. (2021, January 12). Retrieved from https://en.wikipedia.org
