Question

Find the value of the integral `int_0^1 x^2/(1+x^3`𝒅𝒙 using Trapezoidal rule

Answer 1

Let I = `int_0^1 x^2/(1+x^3)dx`

a=0 , b=1
Dividing limits into 4 parts i.e n = 4
`thereforeh=(b-a)/n=1/4=0.25`

𝒙𝟎=0	𝒙𝟏=0.25	𝒙𝟐=0.50	𝒙𝟑=0.75	𝒙𝟒=1.0
𝒚𝟎=0	𝒚𝟏=0.06153	𝒚𝟐=0.2222	𝒚𝟑=0.39560	𝒚𝟒=𝟎.𝟓

Trapezoidal rule :

`"I"=h/2[X+2R]` -----------------(1)

𝑿=𝒔𝒖𝒎 𝒐𝒇 𝒆𝒙𝒕𝒓𝒆𝒎𝒆 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔=𝒚_𝟎+𝒚_𝟒=𝟎+𝟎.𝟓=𝟎.𝟓 𝑹=𝒔𝒖𝒎 𝒐𝒇 𝒓𝒆𝒎𝒂𝒊𝒏𝒊𝒏𝒈 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔 = 𝒚_𝟏+𝒚_𝟐+𝒚_𝟑 =𝟎.𝟎𝟔𝟏𝟓𝟑+𝟎.𝟐𝟐𝟐𝟐+𝟎.𝟑𝟗𝟓𝟔𝟎=𝟎.𝟔𝟕𝟗𝟑𝟑

`"I" = (0.25)/2`(𝟎.𝟓+𝟐(𝟎.𝟑𝟗𝟓𝟔𝟎))  ……………….(from 1)

∴ I = 0.2323