hscishard
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You really want to be female, don't you? LolDo you know many guys who do my subjects?
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Simpson's Rule (1 Viewer)
- Thread starter Riproot
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hscishard
Active Member
Just read cambridge yr 11 3unit
ohexploitable
hey
I'm not a fan of "suppose this then prove it's true".
There's an explanation in Coroneos.
hscishard
Active Member
It does. Translate the graph so that the origin is a+b. If you integrate it normally, it will be (integral a to -a) Ax^3 + (integral a to -a) Bx^2 + Cx + D. Clearly the x^3 is 0 Hence if you use the simpsons rule, it will give an exact value
Lost the site for latex.....
this is from cambridge yr 11 3unit...and yea the proof is kinda weak
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I just realised this is essentially a longer version of the proof in Cambridge lol.
$ at $(x_1,y_1), (x_2, y_2)$ and $(x_3,y_3)$ with $x_1<x_2<x_3$ in such a way that $x_2=\dfrac{x_1+x_3}{2}$ i.e. $x_2$ is the midpoint of $x_1$ and $x_3.\\\\$ Area under parabola $ = \displaystyle\int_{x_1}^{x_3}(Ax^2+Bx+C)\,dx\\\\=\dfrac{A}{3}(x_3^3-x_1^3)+\dfrac{B}{2}(x_3^2-x_1^2)+C(x_3-x_1)\\\\=\dfrac{x_3-x_1}{6}\left[2A(x_3^2+x_1x_3+x_1^2)+3B(x_3+x_1)+6C\right])
^2+B\left(\dfrac{x_1+x_3}{2}\right)+C\\\\y_2=\dfrac{A}{4}(x_1^2+2x_1x_3+x_3^2)+\dfrac{B}{2}(x_1+x_3)+C\\\\4y_2=A(x_1^2+2x_1x_3+x_3^2)+2B(x_1+x_3)+4C\\\\$We now have $\\\\y_1=Ax_1^2+Bx_1+C\\\\4y_2=A(x_1^2+2x_1x_3+x_3^2)+2B(x_1+x_3)+4C\\\\y_3=Ax_3^2+Bx_3+C\\\\\Rightarrow y_1+4y_2+y_3=2A(x_1^2+x_1x_3+x_3^2)+3B(x_1+x_2)+6C\\\\\therefore $ Area under parabola $=\dfrac{x_3-x_1}{6}\left[y_1+4y_2+y_3]\right \approx \displaystyle\int_{x_1}^{x_3}f(x)\,dx )
If you let a = x1 and b = x3 then you get the desired result.
Repeated application of this formula to beyond two intervals leads to the generalised formula for Simpson's rule.
If you let a = x1 and b = x3 then you get the desired result.
Repeated application of this formula to beyond two intervals leads to the generalised formula for Simpson's rule.
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ohexploitable
hey
ohexploitable
hey
See my post above.
ohexploitable
hey
@ Trebla,
Wow, that's really neat, thanks
Wow, that's really neat, thanks
HOW DO YOU NOT KNOW LEIGHTON MEESTER?!<<< is that you in the pic, I thought you were a guy
I GONNA BETCHSLAP THE DUMB OUTTA YOU!
I mean... I'm just... what?
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ohexploitable
hey
*Leighton
I was typing quickly and angrily!*Leighton
hscishard
Active Member
Are you serious?
About what?
hscishard
Active Member
typin quickly and angrily
MrBrightside
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yeah agreed, my teacher teaches lessons really fast, but the pace will go is really slow (topic wise).
Topic wise we're fine, in class = too slow for my awesomeness
Topic wise we're fine, in class = too slow for my awesomeness