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Compute the Value of ∫ π 2 0 √ Sin X + Cos X D X Using Simpson’S (1/3)Rd Rule by Dividing into Six Subintervals. - Applied Mathematics 2

Sum

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

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Solution

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=𝒚𝟎+𝒚𝟔=𝟏+𝟏=𝟐
𝑬=𝒔𝒖𝒎 𝒐𝒇 𝒆𝒗𝒆𝒏 𝒃𝒂𝒔𝒆 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔= 𝒚𝟐+𝒚𝟒=𝟐.𝟑𝟑𝟕𝟔
𝑶=𝒔𝒖𝒎 𝒐𝒇 𝒐𝒅𝒅 𝒃𝒂𝒔𝒆 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔= 𝒚𝟏+𝒚𝟑+𝒚𝟒=𝟑.𝟒𝟎𝟐𝟔

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

I = 1.7693

Concept: Numerical Integration‐ by Simpson’S 1/3rd
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