Trapezoidal rule 4

Evaluate

By (I) Trapezoidal rule ,  (ii) Simpson's one - third rule, correct to four decimals taking 

h = 0.1

Solution :-

h=0.1 

Y = f(x) 

X                   Y

0                    0

0.1                 0.00999

0.2                 0.03998

0.3                 0.08987

0.4                 0.15932

0.5                 0.24740

0.6                 0.35227

0.7                 0.47063

0.8                 0.59719

0.9                 0.72428

1.0                 0.84147

(I) Using Trapezoidal rule

Integration of y dx = h/2 [( y0 + yn) + 2(y1 + y2 +.... ...... + y(n-1) ]

=> 0.1/2[(0 + 0.84147) + 2(0.00999 + 0.03998 + 0.08987 + 0.15932 + 0.24740+ 0.35227 + 0.47063 + 0.59719 + 0.72428) ]

=> 0.05 [ 0.84147 + 2× 2.69093 ]

=> 0.05 [ 0.84147 + 5.38186 ]

=> 0.05 × 6.22333

=> 0.3111665

(ii)  Using Simpson's 1/3rd rule. 

Integration of y dx = h/3 [( y0 + yn) + 4(y1 + y3 +.... ...... + y(n-1) + 2(y2+ y4 + .......... + y(n - 2) ]

=> 0.1/3 [ ( 0 + 0.84147) + 2(0.00999 + 0.08987 +  0.24740+ 0.47063  + 0.72428) + 4(0.03998  + 0.15932 +  0.35227 + 0.59719 ) ]

=> 0.333 [ 0.84147 + 4 × 1.54217+ 2 × 1.14876]

=> 0.333 [ 0.84147 + 6.16868 + 2.29752 ]

=> 0.333 × 9.30767

=> 0.039945411