Numerical Integration | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
4. Numerical Integration 
4.1 Find the value of the integral 
? ?
?? ?? ???? ?? ????
??????? 
by using Simpson's 
?? ?? -rule correct up to 4 decimal places. Take 8 sub-intervals in 
your computation. 
(2010 : 20 Marks) 
Solution: 
Analysis and computer programming: 
Given, 
 Integral ?=? ?
5
1=
?log
10
??????? ?? ?=log
10
??? 
 
?? =0.5= length of sub interval 
By Simpson's 1/3- rule, 
? ?
5
1
?log
10
??????? ?=
?? 3
[(?? 0
+?? 8
)+2(?? 2
+?? 4
+?? 6
)+4(?? 1
+?? 3
+?? 5
+?? 7
)]
?=
?? 3
[(0+0.699)+2(0.3010+0.4771+0.6021)+4(0.1761+0.3979+0.5441+0.6532)]
?=1.7574 (approx.) 
 
4.2 Calculate ?
?? ????
?
????
?? +?? (upto 3 places of decimal) by dividing the range into 8 equal 
parts by Simpson's ?? /?? rd 
 rule. 
(2011 : 12 Marks) 
Solution: 
Page 2


Edurev123 
4. Numerical Integration 
4.1 Find the value of the integral 
? ?
?? ?? ???? ?? ????
??????? 
by using Simpson's 
?? ?? -rule correct up to 4 decimal places. Take 8 sub-intervals in 
your computation. 
(2010 : 20 Marks) 
Solution: 
Analysis and computer programming: 
Given, 
 Integral ?=? ?
5
1=
?log
10
??????? ?? ?=log
10
??? 
 
?? =0.5= length of sub interval 
By Simpson's 1/3- rule, 
? ?
5
1
?log
10
??????? ?=
?? 3
[(?? 0
+?? 8
)+2(?? 2
+?? 4
+?? 6
)+4(?? 1
+?? 3
+?? 5
+?? 7
)]
?=
?? 3
[(0+0.699)+2(0.3010+0.4771+0.6021)+4(0.1761+0.3979+0.5441+0.6532)]
?=1.7574 (approx.) 
 
4.2 Calculate ?
?? ????
?
????
?? +?? (upto 3 places of decimal) by dividing the range into 8 equal 
parts by Simpson's ?? /?? rd 
 rule. 
(2011 : 12 Marks) 
Solution: 
 
Here the range of integration is (2,10) . 
Dividing it into 8 equal parts each of width 
10-2
8
=
8
8
=1, we get, h=1. 
By Simpson's 1/3 rule, we get 
? ?
10
2
?
????
1+?? ?=
h
3
[?? 0
+?? 8
+4(?? 1
+?? 3
+?? 5
+?? 7
)+2(?? 2
+?? 4
+?? 6
)]
?=
1
3
[
1
3
+
1
11
+4(
1
4
+
1
6
+
1
8
+
1
10
)+2(
1
5
+
1
7
+
1
9
)]
?=
1
3
×3.8988453
?=1.2996151=1.300
 
4.3 A solid of revolution is formed by rotating about the ?? -axis, the area between 
the ?? -axis, the line ?? =?? and ?? =?? and a curve through the points with the 
following co-ordinates : 
 
Find the volume of the solid. 
(2011: 20 marks) 
Solution: 
Page 3


Edurev123 
4. Numerical Integration 
4.1 Find the value of the integral 
? ?
?? ?? ???? ?? ????
??????? 
by using Simpson's 
?? ?? -rule correct up to 4 decimal places. Take 8 sub-intervals in 
your computation. 
(2010 : 20 Marks) 
Solution: 
Analysis and computer programming: 
Given, 
 Integral ?=? ?
5
1=
?log
10
??????? ?? ?=log
10
??? 
 
?? =0.5= length of sub interval 
By Simpson's 1/3- rule, 
? ?
5
1
?log
10
??????? ?=
?? 3
[(?? 0
+?? 8
)+2(?? 2
+?? 4
+?? 6
)+4(?? 1
+?? 3
+?? 5
+?? 7
)]
?=
?? 3
[(0+0.699)+2(0.3010+0.4771+0.6021)+4(0.1761+0.3979+0.5441+0.6532)]
?=1.7574 (approx.) 
 
4.2 Calculate ?
?? ????
?
????
?? +?? (upto 3 places of decimal) by dividing the range into 8 equal 
parts by Simpson's ?? /?? rd 
 rule. 
(2011 : 12 Marks) 
Solution: 
 
Here the range of integration is (2,10) . 
Dividing it into 8 equal parts each of width 
10-2
8
=
8
8
=1, we get, h=1. 
By Simpson's 1/3 rule, we get 
? ?
10
2
?
????
1+?? ?=
h
3
[?? 0
+?? 8
+4(?? 1
+?? 3
+?? 5
+?? 7
)+2(?? 2
+?? 4
+?? 6
)]
?=
1
3
[
1
3
+
1
11
+4(
1
4
+
1
6
+
1
8
+
1
10
)+2(
1
5
+
1
7
+
1
9
)]
?=
1
3
×3.8988453
?=1.2996151=1.300
 
4.3 A solid of revolution is formed by rotating about the ?? -axis, the area between 
the ?? -axis, the line ?? =?? and ?? =?? and a curve through the points with the 
following co-ordinates : 
 
Find the volume of the solid. 
(2011: 20 marks) 
Solution: 
 
The required volume is 
=?? ? ?
1
0
?? 2
???? =?? ? ?
1
0
?????? 
By Simpson's 1/3 rd rule, 
? ?
1
0
??????? ?=
h
3
[?? 0
+?? 4
+4(?? 1
+?? 3
)+2?? 2
]
?=
0.2
3
[(1+0.708)+4(0.9793+0.8261)+2×0.9195]?[?h=
1-0
5
=0.2]
?=0.3570
?? Required volume ?=?? ×? ?
1
0
???????
?=3.14×0.3570=1.12098 units cube 
 
4.4 The velocity of a train which starts from rest is given in the following table. The 
time is in minutes and velocity is in ???? /???? . 
?? 2 4 6 8 10 12 14 16 18 20 
?? 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 
 
Estimate approximately the total distance run in 30 minutes by using composite 
Simpson's ?? /?? rules. 
(2013 : 15 Marks) 
Solution: 
The distance travelled by the train is given by 
?? =??
?? 0
?????? 
where ?? is the velocity at any time ?? . We estimate this integral by Simpson's rule. 
First converting the time to hours 
Page 4


Edurev123 
4. Numerical Integration 
4.1 Find the value of the integral 
? ?
?? ?? ???? ?? ????
??????? 
by using Simpson's 
?? ?? -rule correct up to 4 decimal places. Take 8 sub-intervals in 
your computation. 
(2010 : 20 Marks) 
Solution: 
Analysis and computer programming: 
Given, 
 Integral ?=? ?
5
1=
?log
10
??????? ?? ?=log
10
??? 
 
?? =0.5= length of sub interval 
By Simpson's 1/3- rule, 
? ?
5
1
?log
10
??????? ?=
?? 3
[(?? 0
+?? 8
)+2(?? 2
+?? 4
+?? 6
)+4(?? 1
+?? 3
+?? 5
+?? 7
)]
?=
?? 3
[(0+0.699)+2(0.3010+0.4771+0.6021)+4(0.1761+0.3979+0.5441+0.6532)]
?=1.7574 (approx.) 
 
4.2 Calculate ?
?? ????
?
????
?? +?? (upto 3 places of decimal) by dividing the range into 8 equal 
parts by Simpson's ?? /?? rd 
 rule. 
(2011 : 12 Marks) 
Solution: 
 
Here the range of integration is (2,10) . 
Dividing it into 8 equal parts each of width 
10-2
8
=
8
8
=1, we get, h=1. 
By Simpson's 1/3 rule, we get 
? ?
10
2
?
????
1+?? ?=
h
3
[?? 0
+?? 8
+4(?? 1
+?? 3
+?? 5
+?? 7
)+2(?? 2
+?? 4
+?? 6
)]
?=
1
3
[
1
3
+
1
11
+4(
1
4
+
1
6
+
1
8
+
1
10
)+2(
1
5
+
1
7
+
1
9
)]
?=
1
3
×3.8988453
?=1.2996151=1.300
 
4.3 A solid of revolution is formed by rotating about the ?? -axis, the area between 
the ?? -axis, the line ?? =?? and ?? =?? and a curve through the points with the 
following co-ordinates : 
 
Find the volume of the solid. 
(2011: 20 marks) 
Solution: 
 
The required volume is 
=?? ? ?
1
0
?? 2
???? =?? ? ?
1
0
?????? 
By Simpson's 1/3 rd rule, 
? ?
1
0
??????? ?=
h
3
[?? 0
+?? 4
+4(?? 1
+?? 3
)+2?? 2
]
?=
0.2
3
[(1+0.708)+4(0.9793+0.8261)+2×0.9195]?[?h=
1-0
5
=0.2]
?=0.3570
?? Required volume ?=?? ×? ?
1
0
???????
?=3.14×0.3570=1.12098 units cube 
 
4.4 The velocity of a train which starts from rest is given in the following table. The 
time is in minutes and velocity is in ???? /???? . 
?? 2 4 6 8 10 12 14 16 18 20 
?? 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 
 
Estimate approximately the total distance run in 30 minutes by using composite 
Simpson's ?? /?? rules. 
(2013 : 15 Marks) 
Solution: 
The distance travelled by the train is given by 
?? =??
?? 0
?????? 
where ?? is the velocity at any time ?? . We estimate this integral by Simpson's rule. 
First converting the time to hours 
h?=2 min. =
2
60
=0.033 hrs. 
?? ?=? ?
20
0
??????? =? ?
4/60
0
??????? +? ?
8/60
4/60
??????? +? ?
12/60
8/60
??????? +? ?
16/60
12/60
??????? +? ?
20/60
16/60
??????? ?=
h
3
[?? 0
+4?? 1
+?? 2
]+
h
3
[?? 2
+4?? 3
+?? 4
]+
h
3
[?? 7
+4?? 8
+?? 9
]
 
where ?? ?? denotes the velocity at the ?? th 
 serial number, i.e., 
?? 0 1 2 3 4 5 6 7 8 9 
?? ?? 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 
 
???? ?=
h
3
[(?? 0
+?? 9
)+2(?? 2
+?? 4
+?? 6
+?? 8
)+4(?? 1
+?? 3
+?? 5
+?? 7
)]
?=
2
60×3
[(16+0)+2(40+51.2+17.6+3.2)+4(28.8+46.4+32.0+8)]=7.78667kms
 
4.5 Draw a flowchart for Simpson's one-third rule. 
(2014: 15 marks) 
Solution: 
 
Page 5


Edurev123 
4. Numerical Integration 
4.1 Find the value of the integral 
? ?
?? ?? ???? ?? ????
??????? 
by using Simpson's 
?? ?? -rule correct up to 4 decimal places. Take 8 sub-intervals in 
your computation. 
(2010 : 20 Marks) 
Solution: 
Analysis and computer programming: 
Given, 
 Integral ?=? ?
5
1=
?log
10
??????? ?? ?=log
10
??? 
 
?? =0.5= length of sub interval 
By Simpson's 1/3- rule, 
? ?
5
1
?log
10
??????? ?=
?? 3
[(?? 0
+?? 8
)+2(?? 2
+?? 4
+?? 6
)+4(?? 1
+?? 3
+?? 5
+?? 7
)]
?=
?? 3
[(0+0.699)+2(0.3010+0.4771+0.6021)+4(0.1761+0.3979+0.5441+0.6532)]
?=1.7574 (approx.) 
 
4.2 Calculate ?
?? ????
?
????
?? +?? (upto 3 places of decimal) by dividing the range into 8 equal 
parts by Simpson's ?? /?? rd 
 rule. 
(2011 : 12 Marks) 
Solution: 
 
Here the range of integration is (2,10) . 
Dividing it into 8 equal parts each of width 
10-2
8
=
8
8
=1, we get, h=1. 
By Simpson's 1/3 rule, we get 
? ?
10
2
?
????
1+?? ?=
h
3
[?? 0
+?? 8
+4(?? 1
+?? 3
+?? 5
+?? 7
)+2(?? 2
+?? 4
+?? 6
)]
?=
1
3
[
1
3
+
1
11
+4(
1
4
+
1
6
+
1
8
+
1
10
)+2(
1
5
+
1
7
+
1
9
)]
?=
1
3
×3.8988453
?=1.2996151=1.300
 
4.3 A solid of revolution is formed by rotating about the ?? -axis, the area between 
the ?? -axis, the line ?? =?? and ?? =?? and a curve through the points with the 
following co-ordinates : 
 
Find the volume of the solid. 
(2011: 20 marks) 
Solution: 
 
The required volume is 
=?? ? ?
1
0
?? 2
???? =?? ? ?
1
0
?????? 
By Simpson's 1/3 rd rule, 
? ?
1
0
??????? ?=
h
3
[?? 0
+?? 4
+4(?? 1
+?? 3
)+2?? 2
]
?=
0.2
3
[(1+0.708)+4(0.9793+0.8261)+2×0.9195]?[?h=
1-0
5
=0.2]
?=0.3570
?? Required volume ?=?? ×? ?
1
0
???????
?=3.14×0.3570=1.12098 units cube 
 
4.4 The velocity of a train which starts from rest is given in the following table. The 
time is in minutes and velocity is in ???? /???? . 
?? 2 4 6 8 10 12 14 16 18 20 
?? 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 
 
Estimate approximately the total distance run in 30 minutes by using composite 
Simpson's ?? /?? rules. 
(2013 : 15 Marks) 
Solution: 
The distance travelled by the train is given by 
?? =??
?? 0
?????? 
where ?? is the velocity at any time ?? . We estimate this integral by Simpson's rule. 
First converting the time to hours 
h?=2 min. =
2
60
=0.033 hrs. 
?? ?=? ?
20
0
??????? =? ?
4/60
0
??????? +? ?
8/60
4/60
??????? +? ?
12/60
8/60
??????? +? ?
16/60
12/60
??????? +? ?
20/60
16/60
??????? ?=
h
3
[?? 0
+4?? 1
+?? 2
]+
h
3
[?? 2
+4?? 3
+?? 4
]+
h
3
[?? 7
+4?? 8
+?? 9
]
 
where ?? ?? denotes the velocity at the ?? th 
 serial number, i.e., 
?? 0 1 2 3 4 5 6 7 8 9 
?? ?? 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 
 
???? ?=
h
3
[(?? 0
+?? 9
)+2(?? 2
+?? 4
+?? 6
+?? 8
)+4(?? 1
+?? 3
+?? 5
+?? 7
)]
?=
2
60×3
[(16+0)+2(40+51.2+17.6+3.2)+4(28.8+46.4+32.0+8)]=7.78667kms
 
4.5 Draw a flowchart for Simpson's one-third rule. 
(2014: 15 marks) 
Solution: 
 
4.6 Use five subintervals to integrate ?
?? ?? ?
????
?? +?? ?? using trapezoidal rule. 
(2014 : 10 Marks) 
Solution: 
Here 
?? (?? )=
1
1+?? 2
?? =0,?? =1, and ?? =5
???h=
1-0
5
=
1
5
=0.5
 
 
Using trapezoidal rule we get 
?? =? ?
1
0
????
1+?? 2
=
h
2
[(?? 0
+?? 5
)+2(?? 1
+?? 2
+?? 4
)] 
?=
0.2
2
[(1.000000+.500000)+2(0.961538+0.862069+0.735294+0.609756)]
?=0.7837314
?? ?=0.78373, correct to five significant figures. 
 The exact value ?=? ?
1
0
?
1
1+?? 2
???? =[tan
-1
??? ]
0
1
?=tan
-1
?1-tan
-1
?0=
?? 4
=0.7853981
? ?
1
0
?
1
1+?? 2
?=0.78540
 
correct to five significant figures. 
??. The error is =0.78540-0.78373=0.00167 
?? Absolute error =0.00167 
4.7 For an integral ?
?? ?? ??? (?? )???? , show that the two-point Gauss quadrative rule is 
given by ?
-?? ?? ??? (?? )???? = ?? (
?? v?? )+?? (
-?? v?? ) . Using this rule, estimate ?
?? ?? ??? ?? ?? ?? ???? . 
(2016: 15 marks) 
Solution: