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Question
Evaluate `int_0^6 dx/(1+3x)`by using 1} Trapezoidal 2} Simpsons (1/3) rd. and 3} Simpsons (3/8) Th rule.
Solution
| X | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Y | 1 | 0.25 | 0.1428 | 0.1 | 0.0769 | 0.0625 | 0.0526 |
| Ordinate | `y_0` | `y_1` | `y_2` | `y_3` | `y_4` | `y_5` | `y_6` |
Trapezoidal Rule: `I=h/2(x+2R)`
X = Sum of extreme value = 1 + 0.0526 = 1.0526
R = Sum of Remaining values = 0.25 + 0.1428 + 0.1 + 0.0769 + 0.0625 = 0.6322
`I=1/2(1.0526+2(0.6322))`
`I=1.1585`
`"Simpsons (1/3) rd rule" `
`I=h/3 (X+2E+40)`
X = Sum of Extreme values =1 + 0.0526 = 1.0526
E = Sum of even ordinates = 0.1428 + 0.0769 = 0.2197
O = Sum of odd ordinates = 0.25 + 0.1 + 0.0625 = 0.4125
`I=1/3 (1.0526+2(0.2197)+4(0.4125)) `
`I=0.5616`
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