Prove that both trapezoid rule and Simpson's rule give an estimate within 0.005 of true value of integral?

- unimatix

- katieb

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- unimatix

Using Maple I've found:
trapezoid rule gives 2.432066146
Simpson's rule gives 2.430797145

- unimatix

\[\int\limits_{a}^{b}\frac{ x^2 }{ \sqrt{(1-x^2)(k^2-x^2)} }\]
a = 3
b = 5
k = 2

- unimatix

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## More answers

- unimatix

Even if someone just has an idea that might be helpful.

- dan815

error bound on simpsons rule

- anonymous

you want to confirm this numerically or you want to prove it using the error bound theorem

- dan815

i think to find error bounds for trapezoidal one

- dan815

take all right end points and all left end points

- dan815

the true value is between those 2 answers

- dan815

if you show that the difference between trap right end points trap left end points method is less than 0.005 then your answer is within 0.005 of the true value

- unimatix

Okay thanks I'll try and see if I can figure out how to do that.

- dan815

I am not completely sure about this argument though

- dan815

it could be possible that both the right end points and the left end points are both greater than the true value, or both less than the true value

- dan815

yeah throw that method out of the window, there is a more mathematical way

- dan815

https://www.youtube.com/watch?v=qVXIU6mKank
This tells you about simpsons error bound

- unimatix

Hah that's funny I just found that. I don't see one for the Trapezoid rule though :-/

- dan815

isnt simpsons just a higher order trap

- unimatix

I don't really know what that means.

- dan815

simpsons rule is just like beight a bit smarter about the trapezoidal rule

- dan815

it will take the average of the areas of the trapezoids in a smarter way

- unimatix

Okay. So it will be found in a similar way?

- dan815

yeah u should understand the pictures of these operations that will make it a lot clearer

- dan815

https://www.youtube.com/watch?v=uc4xJsi99bk

- dan815

ok i see now

- dan815

basically for trapezoidal

- dan815

|dw:1442549451279:dw|

- dan815

|dw:1442549479549:dw|

- dan815

we went from line connections to parabollic connections

- dan815

for trap to simpsons

- dan815

u can keep going on in this way

- dan815

like higher order simpsons rule, instead of parabola u can do x^3 , x^4,x^5 curve connections

- unimatix

Okay that makes sense

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