# Difference between revisions of "Scilab/C4/Integration/English"

Title of script: Numerical Methods for Integration

Author: Shamika

Keywords: Integration, Numerical Methods, integral

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 4-System 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 Rule-I 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*i-1)

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=1-x^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@spoken-tutorial.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.