EXAMPLES


3.4
( i )
 ò f(x) dx
1.8
x =
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
f(x) =
6.050
7.389
9.025
11.023
13.464
16.445
20.086
24.533
29.964

h = (b-a)/(n-1) = (3.4 - 1.8)/(9 -1) = 0.2

 Case(i) : Trapezoidal rule

 
xn
Trapezoidal formula  = 
 ò f(x) dx = h/2 ( f0 + 2(f1 + . . .  + fn-1) + fn )
x0
= 0.2/2 [6.050 + 2( 7.389 + 9.025 + 11.023 + 13.464 + 16.445 + 20.086 + 24.533 ) + 29.964]
= 23.9944
 
Total Error = - (b-a)/12 h2 f iv(x)  
 for some x0 < x < xn
 
= -(3.4 - 1.8)/12 * 0.22 f iv(x)

= -0.00533f iv(x)


Case(ii) : Simpson's 1/3rd rule

Simpson's 1/3rd formula = 
h
 ( f0 + 4f1 + 2f2   + . . . +  2fn-2 + 4fn-1 + fn )
3
= 0.2/3 [6.050 + 4 * 7.389 + 2 * 9.025 + 4 * 11.023 + 2 * 13.464 + 4 * 16.445 
+ 2 * 20.086 + 4 * 24.533 + 29.964]
= 23.91493
Total Error = - 1/90 h5 f iv(x)
  
 for some x0 < x < xn
 
= -1/90 * 0.25 f iv(x)

= 0

Case(iii) : Simpson's 3/8th rule
 Simpson's 3/8th formula = 
3h
 ( f0 + 3f1 + 3f2   + . . . +  2fn-3 + 3fn-2 + 3fn-1 +  fn )
8
= (3* 0.2)/8 [6.050 + 3* 7.389 + 3* 9.025 + 2* 11.023 + 3* 13.464 + 3* 16.445 + 20.086]
+ 0.2/2[20.086 + 2 * 24.533 + 29.964]
= 23.94793
 Total Error = - 3/80 h5 f iv(x1)
  
 for some x0 < x < xn
 
 = -3/80 * 0.25 f iv(x)

= 0

1.0
( ii )
 ò f(x) dx
1.8
x =
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
f(x) =
1.543
1.669
1.811
1.971
2.151
2.352
2.577
2.828
3.107
h = (b-a)/(n-1) = (1.8 - 1.0)/(9 -1) = 0.1

 Case(i) : Trapezoidal rule

 
xn
Trapezoidal formula  = 
 ò f(x) dx = h/2 ( f0 + 2(f1 + . . .  + fn-1) + fn )
x0
= 0.1/2 [1.543 + 2( 1.669 + 1.811 + 1.971 + 2.151 + 2.352 + 2.577 + 2.828 ) + 3.107]
= 1.76840
 
Total Error = - (b-a)/12 h2 f iv(x)  
 for some x0 < x < xn
 
= -(1.8 - 1.0)/12 * 0.12 f iv(x)

= -6.66E-4 f iv(x)


Case(ii) : Simpson's 1/3rd rule

Simpson's 1/3rd formula = 
h
 ( f0 + 4f1 + 2f2   + . . . +  2fn-2 + 4fn-1 + fn )
3
= 0.1/3 [1.543 + 4 * 1.669 + 2 * 1.811 + 4 * 1.971 + 2 * 2.151 + 4 * 2.352 
+ 2 * 2.577 + 4 * 2.828 + 3.107]
= 1.76693
Total Error = - 1/90 h5 f iv(x)
  
 for some x0 < x < xn
 
= -1/90 * 0.15 f iv(x)

= 0

Case(iii) : Simpson's 3/8th rule
 Simpson's 1/3rd formula = 
3h
 ( f0 + 3f1 + 3f2   + . . . +  2fn-3 + 3fn-2 + 3fn-1 +  fn )
8
= (3* 0.2)/8 [1.543 + 3* 1.669 + 3* 1.811 + 2* 1.971 + 3* 2.151 + 3* 2.352 + 2.577]
+ 0.2/2[2.577 + 2 * 2.828 + 3.107]
= 1.76741
 Total Error = - 3/80 h5 f iv(x1)
  
 for some x0 < x < xn
 
 = -3/80 * 0.15 f iv(x)

= 0


 
2
( iii )
 ò f(x) dx
0
x =
0.00
0.12
0.53
0.87
1.08
1.43
2.00
f(x) =
1.0000
0.8869
0.5886
0.4190
0.3396
0.2393
0.1353
h = (b-a)/(n-1) = (2 - 0)/(7 -1) = 0.333

 Case(i) : Trapezoidal rule

 
xn
Trapezoidal formula  = 
 ò f(x) dx = h/2 ( f0 + 2(f1 + . . .  + fn-1) + fn )
x0
= 0.333/2 [1.0000 + 2( 0.8869 + 0.5886 + 0.4190 + 0.3396 + 0.2393  ) + 0.1353]
= 1.01368 
 
Total Error = - (b-a)/12 h2 f iv(x)  
 for some x0 < x < xn
 
= -(2 - 0)/12 * 0.3332 f iv(x)

= -0.0185f iv(x)


Case(ii) : Simpson's 1/3rd rule

Simpson's 1/3rd formula = 
h
 ( f0 + 4f1 + 2f2   + . . . +  2fn-2 + 4fn-1 + fn )
3

= 0.333/3 [1.0000 + 4* 0.8869 + 2* 0.5886 + 4* 0.4190 + 2* 0.3396 + 4* 0.2393 + 0.1353]

= 1.01917 
Total Error = - 1/90 h5 f iv(x)
  
 for some x0 < x < xn
 
= -1/90 * 0.3335 f iv(x)

= 0

Case(iii) : Simpson's 3/8th rule
 Simpson's 1/3rd formula = 
3h
 ( f0 + 3f1 + 3f2   + . . . +  2fn-3 + 3fn-2 + 3fn-1 +  fn )
8
= 0.333/3 [1.0000 + 3* 0.8869 + 3* 0.5886 + 2* 0.4190 + 3* 0.3396 + 3* 0.2393 + 0.1353]

= 1.01706
 

 Total Error = - 3/80 h5 f iv(x1)
  
 for some x0 < x < xn
 
 = -3/80 * 0.3335 f iv(x)

= 0

 

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Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations