1. Find the approximated integral value 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 `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 Simpsons `1/3` Rule



`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))=4.0552`

`4f(x_(1))=4*4.953=19.812`

`2f(x_(2))=2*6.0436=12.0872`

`4f(x_(3))=4*7.3891=29.5564`

`f(x_(4))=9.025`

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

`=0.2/3*(74.5358)`

`=4.9691`

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

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



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

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

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

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

`=4.9691`

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