written 6.1 years ago by | • modified 6.0 years ago |
$ h = \frac { 1-(-1)}{6} = \frac{2}{6} = \frac{1}{3} \\ $
X | -1 | -2/3 | -1/3 | 0 | 1/3 | 2/3 | 1 |
---|---|---|---|---|---|---|---|
Y | 0.5 | 0.6923 | 0.9 | 1 | 0.9 | 0.6923 | 0.5 |
y0 | y1 | y2 | y3 | y4 | y5 | y6 |
$ \text{1) Trapezoidal Rule } \\ $
$ I = \frac{h}{2} ( X + 2R ) = \frac{h}{2} \left[(y_0+y_6) + 2(y_1+y_2+y_3+y_4+y_5) \right ] \\ $
$ = \frac{1}{3*2} \left[(0.5 + 0.5) + 2( 0.6923 + 0.9 + 1 + 0.9 + 0.6923 ) \right ] \\ $
$ = \frac{1}{6} \left[ 9.3692 \right ] = 1.5615 \\ $
$ \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{1}{3*3} [ (0.5 + 0.5) + 2(0.9 + 0.9) + 4(0.6923 + 1 + 0.6923 ) ] \\ $
$ = \frac{1}{9} [ 14. 1384] = 1.5709 \\ $
$ \text{ 3) Simpson's 3/8 th Rule } \\ $
$ I = \frac{3h}{8}[ X +2T + 3R ] \\ $
$ = 3* \frac{1}{3*8} [(y_0 + y_6) + 2(y_3) + 3(y_1 + y_2 + y_4 + y_5) ] \\ $
$ = \frac{1}{8} [ 12.5538 ] = 1.5692 \\ $