Question

Compute the value of `int_0^(pi/2) sqrt(sinx+cosx) dx` using Simpson’s (1/3)^rd rule by dividing into six Subintervals.

Answer 1

Let I = `int_0^(pi/2) sqrt(sinx+cosx) dx` 

Dividing limits into 6 subintervals . n=6

a = 0, `b=pi/2    thereforeh=(b-a)/n=pi/12`

`x_0=0`	`x_1=pi/12`	`x_2=(2pi)/12`	`x_3=(3pi)/12`	`x_4=(4pi)/12`	`x_5=(5pi)/12`	`x_6=(6pi)/12`
`y_0=1`	`y_1=1.1067`	`y_2=1.1688`	`y_3=1.1892`	`y_4=1.1688`	`y_5=1.1067`	`y_6=1`

Simpson’s (𝟏/𝟑)^𝒓𝒅 rule : 

`"I"=h/3[X+2E+40]`  -----------------(1)

𝑿=𝒔𝒖𝒎 𝒐𝒇 𝒆𝒙𝒕𝒓𝒆𝒎𝒆 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔=𝒚_𝟎+𝒚_𝟔=𝟏+𝟏=𝟐
𝑬=𝒔𝒖𝒎 𝒐𝒇 𝒆𝒗𝒆𝒏 𝒃𝒂𝒔𝒆 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔= 𝒚_𝟐+𝒚_𝟒=𝟐.𝟑𝟑𝟕𝟔
𝑶=𝒔𝒖𝒎 𝒐𝒇 𝒐𝒅𝒅 𝒃𝒂𝒔𝒆 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔= 𝒚_𝟏+𝒚_𝟑+𝒚_𝟒=𝟑.𝟒𝟎𝟐𝟔

`"I"=pi/(3xx12)(2+2xx2.3376+4xx3.4026)`……………….(from 1)

I = 1.7693