Visual Cue

Narration

Slide 1

Dear Friends,
Welcome to the Spoken Tutorial on “ Composite Numerical Integration”

Slide 2,3 Learning Objective Slide

At the end of this tutorial, you will learn how to:
 Develop Scilab code for different Composite Numerical Integration algorithms
 Divide the integral into equal intervals
 Apply the algorithm to each interval
 Calculate the composite value of the integral

Slide 4System Requirement slide

* To record this tutorial, I am using
 Ubuntu 12.04 as the operating system
 and Scilab 5.3.3 version

Slide 5 Prerequisites slide

* Before practising this tutorial, a learner should have basic knowledge of
 Scilab and
 Integration using Numerical Methods
 For Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.

Slide 6 Numerical Integration

Numerical Integration is the:
 Study of how the numerical value of an integral can be found
 It is used when exact mathematical integration is not available
 It approximates a definite integral from values of the integrand

Slide 7,8 Composite Trapezoidal RuleI

Let us study Composite Trapezoidal Rule. This rule is
 The extension of trapezoidal rule
 We divide the interval a comma b into n equal intervals
 Then,
 h equal to b minus a divided by n is the common length of the intervals
 Then composite trapezoidal rule is given by
 The integral of the function F of x in the interval a to b is approximately equal to h multiplied by the sum of the values of the function at x zero to x n

Slide 9 Example

Let us solve an example using composite trapezoidal rule.
Assume the number of intervals n is equal to ten.

Switch to Scilab editor
Highlight
function [I1] = Trap_composite(f, a, b, n)
x = linspace(a, b, n+1)
I1 = (h/2)*(2*sum(f(x))  f(x(1))  f(x(n+1)))

Let us look at the code for Composite Trapezoidal Rule on Scilab Editor
 We first define the function with parameters f , a , b , n.
 f refers to the function we have to solve,
 a is the lower limit of the integral,
 b is the upper limit of the integral and
 n is the number of intervals.
 linspace function is used to create ten equal intervals between zero and one
 We find the value of the integral and store it in I one

Click on Execute on Scilab editor and choose Save and Execute the code

Click on Execute on Scilab editor and choose Save and Execute the code.

Switch to Scilab Console
deff ('[y]=f(x)','y=1/(2*x+1)')
Trap_composite(f, 0, 1, 10)

Define the example function by typing:
 d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one by open paranthesis two asterisk x plus one close paranthesis close quote close paranthesis
 Press Enter
 Type
 Trap underscore composite open paranthesis f comma zero comma one comma ten close paranthesis
 Press Enter
 The answer is displayed on the console

Slide 10, 11 Composite Simpson's Rule

Next we shall study Composite simpson's rule. In this rule, we
 decompose the interval a comma b into n is greater than 1 subintervals of equal length
 Apply Simpson's rule to each interval
 We get the value of the integral to be
 h by three multiplied by the sum of f zero, four into f one , two into f two to f n

Slide 12 Example

Let us solve an example using Composite Simpson's rule.
 We are given a function one by one plus x cube d x in the interval one to two
 Let the number of intervals be twenty

Switch to Scilab Editor and show the code for Simp_composite.sci
Highlight
function I = Simp_composite(f, a, b, n)
for i = 1:(n/2)1
x1(i) = x(2*i)
end
for j = 2:n/2
x2(i) = x(2*i1)
end
I = (h/3)*(f(x(1)) + 2*sum(f(x1)) + 4*sum(f(x2)) + f(x(n)))

Let us look at the code for Composite simpson's rule
 We first define the function with parameters f , a , b , n.
 f refers to the function we have to solve,
 a is the lower limit of the integral,
 b is the upper limit of the integral and
 n is the number of intervals.
 We find two sets of points.
 We find the value of the function with one set and multiply it with two
 With the other set, we find the value and multiply it with four
 We sum these values and multiply it with h by three and store the final value in I
Let us execute the code

Click on Execute and choose Save and execute the file Simp_composite.sci

Save and execute the file Simp underscore composite dot s c i

Switch to Scilab Console
Type clc
deff ('[y]=f(x)','y=1/(1+x^3)')
Simp_composite( f, 1, 2 20)

Let me clear the screen first.
Define the function given in the example by typing
 d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one divided by open paranthesis one plus x cube close paranthesis close quote close paranthesis
 Press Enter
 Type Simp underscore composite open paranthesis f comma one comma two comma twenty close paranthesis
 Press Enter
The answer is displayed on the console.

Slide 13, 14 Composite Midpoint Rule

Let us now look at Composite Midpoint Rule.
 It integrates polynomials of degree one or less
 Divides the interval a comma b into n subintervals of equal width
 Finds the midpoint of each interval indicated by x i
 We find the sum of the values of the integral at each midpoint

Slide 15 Example

Let us solve this problem using Composite Midpoint Rule
 We are given a function one minus x square d x in the interval zero to one point five
 We assume n is equal to twenty

Switch to Scilab Editor
Show the file mid_composite.sci
Highlight
function I = mid_composite(f, a, b, n)
x = linspace(a + h/2, b  h/2, n)
I = h*sum(f(x))

Let us look at the code for Composite Midpoint rule
 We first define the function with parameters f , a , b , n.
 f refers to the function we have to solve,
 a is the lower limit of the integral,
 b is the upper limit of the integral and
 n is the number of intervals.
 We find the midpoint of each interval
 Find the value of integral at each midpoint and then find the sum and store it in I.
Let us now solve the example

Click on Execute and choose
Save and execute the file mid_composite.sci

Save and execute the file mid underscore composite dot s c i

On the Scilab Console type: clc
deff ('[y]=f(x)','y=1x^2')
Type mid_composite(f, 0, 1.5, 20)

Let me clear the screen
We define the function given in the example by typing
 d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one minus x square close quote close paranthesis
 Press Enter
 Then type mid underscore composite open paranthesis f comma zero comma one point five comma twenty close paranthesis
 Press Enter
The answer is displayed on the console

Slide 16 Summary

Let us summarize this tutorial. In this tutorial we have learnt to:
 Develop Scilab code for numerical integration
 Find the value of an integral

Show Slide 17
Title: About the Spoken Tutorial Project
 It summarises the Spoken Tutorial project
 If you do not have good bandwidth, you can download and watch it

* Watch the video available at the following link
 It summarises the Spoken Tutorial project
 If you do not have good bandwidth, you can download and watch it

Show Slide 18
Title: Spoken Tutorial Workshops
The Spoken Tutorial Project Team
 Conducts workshops using spoken tutorials
 Gives certificates for those who pass an online test
 For more details, please write to contact@spokentutorial.org

The Spoken Tutorial Project Team
 Conducts workshops using spoken tutorials
 Gives certificates for those who pass an online test
 For more details, please write to contact at spoken hyphen tutorial dot org

Show Slide 12
Title: Acknowledgement
 Spoken Tutorial Project is a part of the Talk to a Teacher project
 It is supported by the National Mission on Education through ICT, MHRD, Government of India
 More information on this Mission is available at

* Spoken Tutorial Project is a part of the Talk to a Teacher project
 It is supported by the National Mission on Education through ICT, MHRD, Government of India
 More information on this Mission is available at
 spoken hyphen tutorial dot org slash NMEICT hyphen Intro


* This is Ashwini Patil signing off. Thank you for joining.
