Subject : Applied Mathematics 2

Topic : Numerical integration

Difficulty : Medium

$$ $$

$ h = \frac { 5.2-4}{6} = 0.2 \\ $

$ \text{1) Trapezoidal Rule } \\ $

$ I = \frac{h}{2} ( X + 2R ) = \frac{0.2}{2} \left[(y_0+y_6) + 2(y_1+y_2+y_3+y_4+y_5) \right ] \\ $

$ = \frac{0.2}{2} \left[(1.3863 +1.6787) + 2( 1.4351 + 1.4816+1.5261+1.5686+1.6094 ) \right ] \\ $

$ = 1.8277 \\ $

$ \text{2) Simpson's 1/3 rd Rule } \\ $

$ I = \frac{h}{3} [ X + 2E + 40 ] \\ $

$ = \frac{h}{3} [ (y_0+ y_6) + 2(y_2+y_4 ) + 4 (y_1 + y_3 + y_5 ) ] \\ $

$ = \frac{0.2}{3} [ (1.3863 + 1.6487) + 2(1.4816 + 1.5686) + 4(1.4351 + 1.5261 + 1.6094) ] \\ $

$ = 1.8278 \\ $

$ \text{ 3) Simpson's 3/8 th Rule } \\ $

$ I = \frac{3h}{8}[ X +2T + 3R ] \\ $

$ = \frac{3 * 0.2}{8} [(y_0 + y_6) + 2(y_3) + 3(y_1 + y_2 + y_4 + y_5) ] \\ $

$ = \frac{3 * 0.2}{8} [(1.3863 + 1.6487) +2(1.5261) + 3(1.4351 + 1.4816 + 1.5686 + 1.6094) ] \\ $

$ =1.8278 \\ $