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)`

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Examples
1. Find Solution 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 `y`

x	1.4	1.6	1.8	2	2.2
y	4.0552	4.953	6.0436	7.3891	9.025

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

`int y dx=(3h)/8 [(y_0+y_4)+2(y_3)+3(y_1+y_2)]`

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

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

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

`int y dx=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 y dx=(3h)/8 [y_0+3y_1+3y_2+2y_3+y_4]`

`y_0=4.0552`

`3y_1=3*4.953=14.859`

`3y_2=3*6.0436=18.1308`

`2y_3=2*7.3891=14.7782`

`y_4=9.025`

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

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

`int y dx=4.5636`

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

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