written 6.2 years ago by | • modified 2.2 years ago |
Subject : Applied Mathematics 2
Topic : Numerical integration
Difficulty : Medium
written 6.2 years ago by | • modified 2.2 years ago |
Subject : Applied Mathematics 2
Topic : Numerical integration
Difficulty : Medium
written 6.1 years ago by | • modified 6.0 years ago |
$$ $$
$ h = \frac { 5.2-4}{6} = 0.2 \\ $
X | 4 | 4.2 | 4.4 | 4.6 | 4.8 | 5 | 5.2 |
---|---|---|---|---|---|---|---|
Y | 1.3863 | 1.4351 | 1.4816 | 1.5261 | 1.5686 | 1.6094 | 1.648 |
y0 | y1 | y2 | y3 | y4 | y5 | y6 |
$ \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 \\ $