Formula

3. Simpsons `3/8` Rule `int y dx = (3h)/8 (y_0 + 2(y_3 + y_6 + ... + y_(n-3)) + 3(y_1 + y_2 + y_4 + y_5 + ... + y_(n-2)+y_(n-1)) + y_n)` `int f(x) dx=(3Delta x )/8 (f(x_(0))+2(f(x_(3))+f(x_(3))+...+f(x_(n-3)))+3(f(x_(1))+f(x_(2))+f(x_(4))+f(x_(5))+...+f(x_(2))+f(x_(n-1)))+f(x_(n)))`

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Examples
1. Find the approximated integral value using Simpson's 3/8 rule

x	f(x)
1.4	4.0552
1.6	4.9530
1.8	6.0436
2.0	7.3891
2.2	9.0250

Solution:
The value of table for `x` and `f(x)`

`x`	`f(x)`
`x_0=1.4`	`f(x_(0))=4.0552`
`x_1=1.6`	`f(x_(1))=4.953`
`x_2=1.8`	`f(x_(2))=6.0436`
`x_3=2`	`f(x_(3))=7.3891`
`x_4=2.2`	`f(x_(4))=9.025`

Method-1:
Using Simpson's `3/8` Rule



`int f(x) dx=(3Delta x )/8 [f(x_(0))+3f(x_(1))+3f(x_(2))+2f(x_(3))+f(x_(4))]`

`f(x_(0))=4.0552`

`3f(x_(1))=3*4.953=14.859`

`3f(x_(2))=3*6.0436=18.1308`

`2f(x_(3))=2*7.3891=14.7782`

`f(x_(4))=9.025`

`int f(x) dx=(3xx0.2)/8 *(4.0552+14.859+18.1308+14.7782+9.025)`

`=(3xx0.2)/8 *(60.8482)`

`=4.5636`

Solution by Simpson's `3/8` Rule is `4.5636`

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Method-2:
Using Simpson's `3/8` Rule



`int f(x) dx=(3Delta x )/8 [(f(x_(0))+f(x_(4)))+2(f(x_(3)))+3(f(x_(1))+f(x_(2)))]`

`=(3xx0.2)/8 [(4.0552 +9.025)+2xx(7.3891)+3xx(4.953+6.0436)]`

`=(3xx0.2)/8 [(4.0552 +9.025)+2xx(7.3891)+3xx(10.9966)]`

`=(3xx0.2)/8 [(13.0802)+(14.7782)+(32.9898)]`

`=4.5636`

Solution by Simpson's `3/8` Rule is `4.5636`

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