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(Created page with '{| Border=1 || Time || Narration |- | 00.01 |Dear Friends, |- | 00.02. | Welcome to the Spoken Tutorial on '''“ Composite Numerical Integration” ''' |- |00.07 |At the …')
 
 
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{| Border=1
 
{| Border=1
  
−
|| Time
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|'''Time'''
−
 
+
|'''Narration'''
−
|| Narration
+
  
 
|-
 
|-
−
| 00.01
+
| 00:01
−
|Dear Friends,  
+
|Dear Friends, Welcome to the Spoken Tutorial on '''Composite Numerical Integration'''.
−
 
+
−
|-
+
−
| 00.02.
+
−
| Welcome to the Spoken Tutorial on '''“ Composite Numerical Integration” '''
+
−
 
+
  
 
|-
 
|-
−
|00.07
+
|00:07
 
|At the end of this tutorial, you will learn how to:  
 
|At the end of this tutorial, you will learn how to:  
  
 
|-
 
|-
−
|00.11
+
|00:11
 
|Develop '''Scilab''' code for different '''Composite Numerical Integration algorithms'''  
 
|Develop '''Scilab''' code for different '''Composite Numerical Integration algorithms'''  
  
 
|-
 
|-
−
| 00.17
+
| 00:17
 
|Divide the '''integral''' into equal '''intervals'''  
 
|Divide the '''integral''' into equal '''intervals'''  
  
 
|-
 
|-
−
|00.21
+
|00:21
 
|Apply the algorithm to each '''interval''' and  
 
|Apply the algorithm to each '''interval''' and  
  
 
|-
 
|-
−
|00.24
+
|00:24
−
|Calculate the '''composite value of the integral'''
+
|Calculate the '''composite value of the integral'''.
  
 
|-
 
|-
−
| 00.28
+
| 00:28
 
|To record this tutorial, I am using  
 
|To record this tutorial, I am using  
  
 
|-
 
|-
−
| 00.30
+
| 00:30
 
| '''Ubuntu 12.04''' as the operating system  
 
| '''Ubuntu 12.04''' as the operating system  
  
 
|-
 
|-
−
|00.34
+
|00:34
−
| with '''Scilab 5.3.3''' version  
+
| with '''Scilab 5.3.3''' version.
  
 
|-
 
|-
−
|00.38
+
|00:38
−
||Before practising this tutorial, a learner should have basic knowledge of  
+
||Before practicing this tutorial, a learner should have basic knowledge of  
  
 
|-
 
|-
−
| 00.42
+
| 00:42
 
|'''Scilab''' and  
 
|'''Scilab''' and  
−
 
  
 
|-
 
|-
−
|00.44
+
|00:44
−
| '''Integration using Numerical Methods'''
+
| '''Integration using Numerical Methods'''.
  
 
|-
 
|-
  
−
| 00.47
+
| 00:47
  
 
| For '''Scilab''', please refer to the relevant tutorials available on the '''Spoken Tutorial''' website.  
 
| For '''Scilab''', please refer to the relevant tutorials available on the '''Spoken Tutorial''' website.  
  
 
|-
 
|-
−
| 00.55
+
| 00:55
−
| '''Numerical Integration''' is the:
+
| '''Numerical Integration''' is the  
  
 
|-
 
|-
−
| 00.58
+
| 00:58
−
| Study of how the numerical value of an '''integral''' can be found  
+
| study of how the numerical value of an '''integral''' can be found.
  
 
|-
 
|-
−
|01.03
+
|01:03
−
| It is used when exact mathematical integration is not available  
+
| It is used when exact mathematical integration is not available.
  
 
|-
 
|-
−
|01.08
+
|01:08
−
|It approximates a definite '''integral''' from values of the '''integrand'''  
+
|It approximates a '''definite integral''' from values of the '''integrand'''.
−
 
+
  
 
|-
 
|-
  
−
|01.15
+
|01:15
  
 
|Let us study '''Composite Trapezoidal Rule.'''  
 
|Let us study '''Composite Trapezoidal Rule.'''  
Line 90: Line 82:
 
|-
 
|-
  
−
|01.18
+
|01:18
  
−
|This rule is the extension of '''trapezoidal rule'''  
+
|This rule is the extension of '''trapezoidal rule'''.
  
 
|-
 
|-
  
−
| 01.22
+
| 01:22
−
|| We divide the interval '''a comma b '''into '''n''' equal intervals  
+
|| We divide the interval '''a comma b ''' into '''n''' equal intervals.
  
 
|-
 
|-
  
−
| 01.29
+
| 01:29
  
−
| Then '''h equal to b minus a divided by n''' is the common length of the intervals  
+
| Then '''h equals to b minus a divided by n''' is the common length of the intervals.
  
 
|-
 
|-
  
−
|01.36
+
|01:36
  
−
| Then '''composite trapezoidal rule''' is given by  
+
| Then '''composite trapezoidal rule''' is given by:
  
 
|-
 
|-
  
−
|01.41
+
|01:41
  
 
|''' 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'''
 
|''' 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'''
Line 119: Line 111:
 
|-
 
|-
  
−
|01.57
+
|01:57
  
 
|| Let us solve an example using '''composite trapezoidal rule.'''  
 
|| Let us solve an example using '''composite trapezoidal rule.'''  
Line 125: Line 117:
 
|-
 
|-
  
−
|02.02
+
|02:02
−
 
+
−
| Assume the number of intervals n is equal to ten.
+
  
 +
| Assume the number of intervals n is equal to ten (n=10).
  
 
|-
 
|-
−
|02.09
+
|02:09
−
|Let us look at the code for '''Composite Trapezoidal Rule''' on '''Scilab Editor'''
+
|Let us look at the code for '''Composite Trapezoidal Rule''' on '''Scilab editor'''
  
 
|-
 
|-
−
| 02.16
+
| 02:16
 
||We first define the function with parameters '''f , a , b , n.'''
 
||We first define the function with parameters '''f , a , b , n.'''
−
 
  
 
|-
 
|-
−
| 02.22
+
| 02:22
 
|'''f '''refers to the function we have to solve,  
 
|'''f '''refers to the function we have to solve,  
 +
 
|-
 
|-
  
−
| 02.25
+
| 02:25
−
|| '''a ''' is the lower limit of the integral,  
+
|| '''a''' is the lower limit of the integral,
 +
 
|-
 
|-
  
−
|02.28
+
|02:28
  
 
||''' b''' is the upper limit of the integral and  
 
||''' b''' is the upper limit of the integral and  
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|-
 
|-
  
−
|02.31
+
|02:31
  
 
| '''n''' is the number of intervals.  
 
| '''n''' is the number of intervals.  
Line 160: Line 152:
 
|-
 
|-
  
−
|02.34
+
|02:34
  
−
| '''linspace''' function is used to create ten equal intervals between zero and one  
+
| '''linspace''' function is used to create ten equal intervals between zero and one.
  
 
|-
 
|-
  
−
| 02.42
+
| 02:42
  
−
|| We find the value of the integral and store it in ''' I one'''
+
|| We find the value of the integral and store it in ''' I one'''.
  
 
|-
 
|-
−
| 02.49
+
| 02:49
−
| 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.  
  
 
|-
 
|-
−
|03.02
+
|03:02
 
|  Define the example function by typing:  
 
|  Define the example function by typing:  
  
 
|-
 
|-
−
| 03.05
+
| 03:05
−
| '''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'''
+
| '''d e f f open parenthesis 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 parenthesis two asterisk x plus one close parenthesis close quote close parenthesis'''
  
 
|-
 
|-
−
| 03.30
+
| 03:30
−
| Press '''Enter '''
+
| Press '''Enter '''. Type '''Trap underscore composite open parenthesis f comma zero comma one comma ten close parenthesis'''  
  
 
|-
 
|-
−
| 03.31
+
|03:41
−
|  Type '''Trap underscore composite open paranthesis f comma zero comma one comma ten close paranthesis
+
| Press '''Enter '''.
−
'''
+
−
|-
+
−
|03.41
+
−
| Press '''Enter '''
+
−
 
+
  
 
|-
 
|-
−
|03.43
+
|03:43
−
| The answer is displayed on the '''console '''
+
| The answer is displayed on the '''console '''.
  
 
|-
 
|-
−
| 03.47
+
| 03:47
−
| Next we shall study '''Composite simpson's rule.'''
+
| Next we shall study '''Composite Simpson's rule.'''
  
 
|-
 
|-
  
−
| 03.51
+
| 03:51
−
 
+
−
| In this rule, we decompose the interval ''' a comma b''' into '''n is greater than 1'''  subintervals of equal length
+
  
 +
| In this rule, we decompose the interval ''' a comma b''' into '''n is greater than 1'''  sub-intervals of equal length.
  
 
|-
 
|-
  
−
| 04.03
+
| 04:03
−
|| Apply '''Simpson's rule''' to each interval  
+
|| Apply '''Simpson's rule''' to each interval.
  
 
|-
 
|-
  
−
| 04.06
+
| 04:06
−
 
+
−
| We get the value of the integral to be
+
  
 +
| We get the value of the integral to be:
  
 
|-
 
|-
  
−
|04.10
+
|04:10
−
 
+
−
| '''h by three multiplied by the sum of f zero, four into f one , two into f two to f n'''
+
  
 +
| '''h by three multiplied by the sum of f zero, four into f one , two into f two to f n'''.
  
 
|-
 
|-
  
−
|04.19
+
|04:19
  
 
||Let us solve an example using '''Composite Simpson's rule. '''
 
||Let us solve an example using '''Composite Simpson's rule. '''
Line 237: Line 221:
 
|-
 
|-
  
−
| 04.24
+
| 04:24
−
 
+
−
|We are given a '''function one by one plus x cube d x in the interval one to two'''
+
  
 +
|We are given a '''function one by one plus x cube d x in the interval one to two'''.
  
 
|-
 
|-
  
−
| 04.32
+
| 04:32
−
 
+
−
| Let the number of intervals be '''twenty '''
+
  
 +
| Let the number of intervals be '''twenty '''.
  
 
|-
 
|-
  
−
|04.37
+
|04:37
−
 
+
−
| Let us look at the code for  '''Composite simpson's rule'''
+
−
 
+
  
 +
| Let us look at the code for  '''Composite Simpson's rule'''.
  
 
|-
 
|-
−
|04.42
+
|04:42
 
| We first define the function with parameters '''f , a , b , n. '''
 
| We first define the function with parameters '''f , a , b , n. '''
  
 
|-
 
|-
  
−
| 04.49
+
| 04:49
  
 
| '''f''' refers to the function we have to solve,  
 
| '''f''' refers to the function we have to solve,  
−
 
−
 
  
 
|-
 
|-
  
−
|04.52
+
|04:52
  
 
||'''a'''  is the lower limit of the integral,  
 
||'''a'''  is the lower limit of the integral,  
−
 
  
 
|-
 
|-
  
−
|04.56
+
|04:56
  
 
| '''b''' is the upper limit of the integral and  
 
| '''b''' is the upper limit of the integral and  
−
 
  
 
|-
 
|-
  
−
| 04.58
+
| 04:58
  
 
| '''n''' is the number of intervals.  
 
| '''n''' is the number of intervals.  
−
 
−
 
  
 
|-
 
|-
  
−
| 05.02
+
| 05:02
  
 
|We find two sets of points.  
 
|We find two sets of points.  
−
 
  
 
|-
 
|-
  
−
| 05.04
+
| 05:04
−
 
+
−
| We find the value of the function with one set and multiply it with two
+
−
 
+
  
 +
| We find the value of the function with one set and multiply it with two.
  
 
|-
 
|-
  
−
| 05.10
+
| 05:10
−
 
+
−
| With the other set, we find the value and multiply it with four
+
−
 
+
  
 +
| With the other set, we find the value and multiply it with four.
  
 
|-
 
|-
  
−
| 05.16
+
| 05:16
  
−
||We sum these values and multiply it with '''h by three and store the final value in I '''
+
||We sum these values and multiply it with '''h by three and store the final value in I '''.
  
 
|-
 
|-
  
−
| 05.24
+
| 05:24
  
−
||Let us execute the code  
+
||Let us execute the code.
  
 
|-
 
|-
  
−
| 05.28
+
| 05:28
  
−
|| Save and execute the file '''Simp underscore composite dot s c i'''
+
|| Save and execute the file '''Simp underscore composite dot s c i'''.
  
 
|-
 
|-
  
−
| 05.39
+
| 05:39
  
 
|Let me clear the screen first.  
 
|Let me clear the screen first.  
Line 340: Line 309:
 
|-
 
|-
  
−
| 05.42
+
| 05:42
  
−
| Define the function given in the example by typing  
+
| Define the function given in the example by typing:
  
 
|-
 
|-
  
−
|05.45
+
|05:45
−
 
+
−
|'''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'''
+
  
 +
|'''d e f f open parenthesis 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 parenthesis one plus x cube close parenthesis close quote close parenthesis'''
  
 
|-
 
|-
  
−
|06.12
+
|06:12
−
 
+
−
| Press '''Enter '''
+
  
 +
| Press '''Enter '''.
  
 
|-
 
|-
  
−
| 06.14
+
| 06:14
−
 
+
−
| Type '''Simp underscore composite open paranthesis f comma one comma two comma twenty close paranthesis'''
+
  
 +
| Type '''Simp underscore composite open parenthesis f comma one comma two comma twenty close parenthesis'''
  
 
|-
 
|-
  
−
|06.24
+
|06:24
−
 
+
−
||Press '''Enter '''
+
−
 
+
  
 +
||Press '''Enter '''.
  
 
|-
 
|-
  
−
| 06.26
+
| 06:26
  
 
| The answer is displayed on the console.  
 
| The answer is displayed on the console.  
−
 
  
 
|-
 
|-
  
−
| 06.31
+
| 06:31
  
 
| Let us now look at '''Composite Midpoint Rule.'''
 
| Let us now look at '''Composite Midpoint Rule.'''
−
 
−
 
  
 
|-
 
|-
  
−
| 06.35
+
| 06:35
  
−
| It integrates polynomials of degree one or less  
+
| It integrates polynomials of degree one or less,
  
 
|-
 
|-
  
−
|06.40
+
|06:40
  
−
| Divides the interval '''a comma b''' into a ''' subintervals'''of equal width  
+
| divides the interval '''a comma b''' into a ''' sub-intervals'''of equal width.
  
 
|-
 
|-
  
−
|06.49
+
|06:49
  
−
| Finds the midpoint of each interval indicated by '''x i '''
+
| Finds the midpoint of each interval indicated by '''x i '''.
  
 
|-
 
|-
  
−
|06.54
+
|06:54
  
−
| We find the sum of the values of the integral at each midpoint  
+
| We find the sum of the values of the integral at each midpoint.
  
 
|-
 
|-
  
−
|07.00
+
|07:00
  
−
| Let us solve this problem using '''Composite Midpoint Rule'''
+
| Let us solve this problem using '''Composite Midpoint Rule'''.
  
 
|-
 
|-
  
−
|07.05
+
|07:05
  
−
| '''We are given a function one minus x square d x in the interval zero to one point five'''
+
| '''We are given a function one minus x square d x in the interval zero to one point five'''.
  
 
|-
 
|-
  
−
|07.15
+
|07:15
  
−
| We assume '''n''' is equal to '''twenty '''
+
| We assume '''n''' is equal to '''twenty '''.
  
 
|-
 
|-
  
−
|07.18
+
|07:18
  
−
| Let us look at the code for '''Composite Midpoint rule'''
+
| Let us look at the code for '''Composite Midpoint rule'''.
  
 
|-
 
|-
  
−
|07.24
+
|07:24
  
 
| We first define the function with parameters '''f , a , b , n. '''
 
| We first define the function with parameters '''f , a , b , n. '''
Line 444: Line 405:
 
|-
 
|-
  
−
|07.30
+
|07:30
  
 
| '''f ''' refers to the function we have to solve,  
 
| '''f ''' refers to the function we have to solve,  
Line 450: Line 411:
 
|-
 
|-
  
−
|07.33
+
|07:33
  
 
| '''a'''  is the lower limit of the integral,  
 
| '''a'''  is the lower limit of the integral,  
Line 456: Line 417:
 
|-
 
|-
  
−
|07.36
+
|07:36
  
 
| '''b ''' is the upper limit of the integral and  
 
| '''b ''' is the upper limit of the integral and  
Line 462: Line 423:
 
|-
 
|-
  
−
|07.39
+
|07:39
  
 
| '''n ''' is the number of intervals.  
 
| '''n ''' is the number of intervals.  
Line 468: Line 429:
 
|-
 
|-
  
−
|07.41
+
|07:41
−
 
+
−
| We find the midpoint of each interval
+
  
 +
| We find the midpoint of each interval.
  
 
|-
 
|-
  
−
|07.45
+
|07:45
  
 
| Find the value of integral at each midpoint and then find the sum and store it in '''I.'''  
 
| Find the value of integral at each midpoint and then find the sum and store it in '''I.'''  
Line 481: Line 441:
 
|-
 
|-
  
−
|07.53
+
|07:53
  
−
| Let us now solve the example  
+
| Let us now solve the example.
  
 
|-
 
|-
  
−
|07.55
+
|07:55
  
−
| Save and execute the file '''mid underscore composite dot s c i'''  
+
| Save and execute the file '''mid underscore composite dot s c i'''.
  
 
|-
 
|-
  
−
|08.04
+
|08:04
  
−
| Let me clear the screen  
+
| Let me clear the screen.
  
 
|-
 
|-
  
−
|08.08
+
|08:08
  
−
| We define the function given in the example by typing  
+
| We define the function given in the example by typing:
  
 
|-
 
|-
  
−
|08.13
+
|08:13
  
−
| '''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'''
+
| '''d e f f open parenthesis 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 parenthesis'''
  
 
|-
 
|-
  
−
|08.37
+
|08:37
  
−
| Press '''Enter'''  
+
| Press '''Enter'''.
  
 
|-
 
|-
  
−
|08.39
+
|08:39
  
−
| Then type '''mid underscore composite open paranthesis f comma zero comma one point five comma twenty close paranthesis'''
+
| Then type '''mid underscore composite open parenthesis f comma zero comma one point five comma twenty close parenthesis'''
  
 
|-
 
|-
  
−
|08.53
+
|08:53
  
−
|Press '''Enter '''
+
|Press '''Enter '''. The answer is displayed on the '''console'''.
−
 
+
−
|-
+
−
 
+
−
|08.54
+
−
 
+
−
| The answer is displayed on the '''console'''
+
  
 
|-
 
|-
  
−
|08.59
+
|08:59
  
 
| Let us summarize this tutorial.  
 
| Let us summarize this tutorial.  
Line 541: Line 495:
 
|-
 
|-
  
−
|09.02
+
|09:02
  
 
| In this tutorial we have learnt to:  
 
| In this tutorial we have learnt to:  
Line 547: Line 501:
 
|-
 
|-
  
−
|09.04
+
|09:04
  
 
| Develop '''Scilab''' code for '''numerical integration'''  
 
| Develop '''Scilab''' code for '''numerical integration'''  
Line 553: Line 507:
 
|-
 
|-
  
−
|09.08
+
|09:08
−
 
+
−
| Find the value of an '''integral'''
+
  
 +
| Find the value of an '''integral'''.
  
 
|-
 
|-
−
|09.11
+
|09:11
  
−
| Watch the video available at the link shown below  
+
| Watch the video available at the link shown below.
  
 
|-
 
|-
  
−
| 09.15
+
| 09:15
−
 
+
−
| It summarises the Spoken Tutorial project
+
−
 
+
  
 +
| It summarizes the Spoken Tutorial project.
  
 
|-
 
|-
  
−
|09.18
+
|09:18
  
−
||If you do not have good bandwidth, you can download and watch it  
+
||If you do not have good bandwidth, you can download and watch it.
  
 
|-
 
|-
  
−
|09.23
+
|09:23
  
−
||The spoken tutorial Team
+
||The spoken tutorial Team:
  
 
|-
 
|-
  
−
|09.25
+
|09:25
  
 
||Conducts workshops using spoken tutorials  
 
||Conducts workshops using spoken tutorials  
−
 
  
 
|-
 
|-
  
−
|06.29
+
|09:29
−
 
+
−
||Gives certificates to those who pass an online test
+
  
 +
||Gives certificates to those who pass an online test.
  
 
|-
 
|-
  
−
|09.32
+
|09:32
−
 
+
−
||For more details, please write to contact@spoken-tutorial.org
+
  
 +
||For more details, please write to contact@spoken-tutorial.org.
  
 
|-
 
|-
  
−
|09.40
+
|09:40
  
−
|Spoken Tutorial Project is a part of the Talk to a Teacher project  
+
|Spoken Tutorial Project is a part of the Talk to a Teacher project.
  
 
|-
 
|-
  
−
| 09.45
+
| 09:45
  
 
| It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.  
 
| It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.  
 
|-
 
|-
  
−
| 09.52
+
| 09:52
  
−
|More information on this mission is available at  http://spoken-tutorial.org/NMEICT-Intro
+
|More information on this mission is available at  http://spoken-tutorial.org/NMEICT-Intro.
  
 
|-
 
|-
  
−
| 10.03
+
| 10:03
  
−
|This is Ashwini Patil signing off. Thank you for joining.
+
|This is Ashwini Patil, signing off. Thank you for joining.

Latest revision as of 11:37, 10 March 2017

Time Narration
00:01 Dear Friends, Welcome to the Spoken Tutorial on Composite Numerical Integration.
00:07 At the end of this tutorial, you will learn how to:
00:11 Develop Scilab code for different Composite Numerical Integration algorithms
00:17 Divide the integral into equal intervals
00:21 Apply the algorithm to each interval and
00:24 Calculate the composite value of the integral.
00:28 To record this tutorial, I am using
00:30 Ubuntu 12.04 as the operating system
00:34 with Scilab 5.3.3 version.
00:38 Before practicing this tutorial, a learner should have basic knowledge of
00:42 Scilab and
00:44 Integration using Numerical Methods.
00:47 For Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.
00:55 Numerical Integration is the
00:58 study of how the numerical value of an integral can be found.
01:03 It is used when exact mathematical integration is not available.
01:08 It approximates a definite integral from values of the integrand.
01:15 Let us study Composite Trapezoidal Rule.
01:18 This rule is the extension of trapezoidal rule.
01:22 We divide the interval a comma b into n equal intervals.
01:29 Then h equals to b minus a divided by n is the common length of the intervals.
01:36 Then composite trapezoidal rule is given by:
01:41 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
01:57 Let us solve an example using composite trapezoidal rule.
02:02 Assume the number of intervals n is equal to ten (n=10).
02:09 Let us look at the code for Composite Trapezoidal Rule on Scilab editor
02:16 We first define the function with parameters f , a , b , n.
02:22 f refers to the function we have to solve,
02:25 a is the lower limit of the integral,
02:28 b is the upper limit of the integral and
02:31 n is the number of intervals.
02:34 linspace function is used to create ten equal intervals between zero and one.
02:42 We find the value of the integral and store it in I one.
02:49 Click on Execute on Scilab editor and choose Save and execute the code.
03:02 Define the example function by typing:
03:05 d e f f open parenthesis 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 parenthesis two asterisk x plus one close parenthesis close quote close parenthesis
03:30 Press Enter . Type Trap underscore composite open parenthesis f comma zero comma one comma ten close parenthesis
03:41 Press Enter .
03:43 The answer is displayed on the console .
03:47 Next we shall study Composite Simpson's rule.
03:51 In this rule, we decompose the interval a comma b into n is greater than 1 sub-intervals of equal length.
04:03 Apply Simpson's rule to each interval.
04:06 We get the value of the integral to be:
04:10 h by three multiplied by the sum of f zero, four into f one , two into f two to f n.
04:19 Let us solve an example using Composite Simpson's rule.
04:24 We are given a function one by one plus x cube d x in the interval one to two.
04:32 Let the number of intervals be twenty .
04:37 Let us look at the code for Composite Simpson's rule.
04:42 We first define the function with parameters f , a , b , n.
04:49 f refers to the function we have to solve,
04:52 a is the lower limit of the integral,
04:56 b is the upper limit of the integral and
04:58 n is the number of intervals.
05:02 We find two sets of points.
05:04 We find the value of the function with one set and multiply it with two.
05:10 With the other set, we find the value and multiply it with four.
05:16 We sum these values and multiply it with h by three and store the final value in I .
05:24 Let us execute the code.
05:28 Save and execute the file Simp underscore composite dot s c i.
05:39 Let me clear the screen first.
05:42 Define the function given in the example by typing:
05:45 d e f f open parenthesis 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 parenthesis one plus x cube close parenthesis close quote close parenthesis
06:12 Press Enter .
06:14 Type Simp underscore composite open parenthesis f comma one comma two comma twenty close parenthesis
06:24 Press Enter .
06:26 The answer is displayed on the console.
06:31 Let us now look at Composite Midpoint Rule.
06:35 It integrates polynomials of degree one or less,
06:40 divides the interval a comma b into a sub-intervalsof equal width.
06:49 Finds the midpoint of each interval indicated by x i .
06:54 We find the sum of the values of the integral at each midpoint.
07:00 Let us solve this problem using Composite Midpoint Rule.
07:05 We are given a function one minus x square d x in the interval zero to one point five.
07:15 We assume n is equal to twenty .
07:18 Let us look at the code for Composite Midpoint rule.
07:24 We first define the function with parameters f , a , b , n.
07:30 f refers to the function we have to solve,
07:33 a is the lower limit of the integral,
07:36 b is the upper limit of the integral and
07:39 n is the number of intervals.
07:41 We find the midpoint of each interval.
07:45 Find the value of integral at each midpoint and then find the sum and store it in I.
07:53 Let us now solve the example.
07:55 Save and execute the file mid underscore composite dot s c i.
08:04 Let me clear the screen.
08:08 We define the function given in the example by typing:
08:13 d e f f open parenthesis 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 parenthesis
08:37 Press Enter.
08:39 Then type mid underscore composite open parenthesis f comma zero comma one point five comma twenty close parenthesis
08:53 Press Enter . The answer is displayed on the console.
08:59 Let us summarize this tutorial.
09:02 In this tutorial we have learnt to:
09:04 Develop Scilab code for numerical integration
09:08 Find the value of an integral.
09:11 Watch the video available at the link shown below.
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09:18 If you do not have good bandwidth, you can download and watch it.
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10:03 This is Ashwini Patil, signing off. Thank you for joining.

Contributors and Content Editors

Gaurav, PoojaMoolya, Sandhya.np14

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