Formula

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

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Examples
1. Find Solution using Simpson's 1/3 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 Simpsons `1/3` Rule

`int y dx=h/3 [(y_0+y_4)+4(y_1+y_3)+2(y_2)]`

`int y dx=0.2/3 [(4.0552 +9.025)+4xx(4.953+7.3891)+2xx(6.0436)]`

`int y dx=0.2/3 [(4.0552 +9.025)+4xx(12.3421)+2xx(6.0436)]`

`int y dx=0.2/3 [(13.0802)+(49.3684)+(12.0872)]`

`int y dx=4.9691`

Solution by Simpson's `1/3` Rule is `4.9691`

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Method-2:
Using Simpsons `1/3` Rule

`int y dx=h/3 [y_0+4y_1+2y_2+4y_3+y_4]`

`y_0=4.0552`

`4y_1=4*4.953=19.812`

`2y_2=2*6.0436=12.0872`

`4y_3=4*7.3891=29.5564`

`y_4=9.025`

`int y dx=0.2/3*(4.0552+19.812+12.0872+29.5564+9.025)`

`int y dx=0.2/3*(74.5358)`

`int y dx=4.9691`

Solution by Simpson's `1/3` Rule is `4.9691`

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