Find the approximated integral value using Simpson's 1/3 rule
| x | f(x) |
| 0.0 | 1.0000 |
| 0.1 | 0.9975 |
| 0.2 | 0.9900 |
| 0.3 | 0.9776 |
| 0.4 | 0.8604 |
Solution:The value of table for `x` and `f(x)`
| `x` | `f(x)` |
| `x_0=0` | `f(x_(0))=1` |
| `x_1=0.1` | `f(x_(1))=0.9975` |
| `x_2=0.2` | `f(x_(2))=0.99` |
| `x_3=0.3` | `f(x_(3))=0.9776` |
| `x_4=0.4` | `f(x_(4))=0.8604` |
Method-1:Using Simpsons `1/3` Rule
`int f(x) dx=(Delta x )/3 (f(x_(0))+4(f(x_(1))+f(x_(3))+f(x_(5))+...+f(x_(n-1)))+2(f(x_(2))+f(x_(4))+f(x_(6))+...+f(x_(n-2)))+f(x_(n)))`
`int f(x) dx=(Delta x )/3 [f(x_(0))+4f(x_(1))+2f(x_(2))+4f(x_(3))+f(x_(4))]`
`f(x_(0))=1`
`4f(x_(1))=4*0.9975=3.99`
`2f(x_(2))=2*0.99=1.98`
`4f(x_(3))=4*0.9776=3.9104`
`f(x_(4))=0.8604`
`int f(x) dx=0.1/3*(1+3.99+1.98+3.9104+0.8604)`
`=0.1/3*(11.7408)`
`=0.3914`
Solution by Simpson's `1/3` Rule is `0.3914`
Method-2:Using Simpsons `1/3` Rule
`int f(x) dx=(Delta x )/3 (f(x_(0))+4(f(x_(1))+f(x_(3))+f(x_(5))+...+f(x_(n-1)))+2(f(x_(2))+f(x_(4))+f(x_(6))+...+f(x_(n-2)))+f(x_(n)))`
`int f(x) dx=(Delta x )/3 [(f(x_(0))+f(x_(4)))+4(f(x_(1))+f(x_(3)))+2(f(x_(2)))]`
`=0.1/3 [(1 +0.8604)+4xx(0.9975+0.9776)+2xx(0.99)]`
`=0.1/3 [(1 +0.8604)+4xx(1.9751)+2xx(0.99)]`
`=0.1/3 [(1.8604)+(7.9004)+(1.98)]`
`=0.3914`
Solution by Simpson's `1/3` Rule is `0.3914`
This material is intended as a summary. Use your textbook for detail explanation.
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