## unimatix one year ago Prove that both trapezoid rule and Simpson's rule give an estimate within 0.005 of true value of integral?

1. unimatix

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

2. unimatix

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

3. unimatix

@SolomonZelman @pooja195

4. unimatix

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

5. dan815

error bound on simpsons rule

6. anonymous

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

7. dan815

i think to find error bounds for trapezoidal one

8. dan815

take all right end points and all left end points

9. dan815

the true value is between those 2 answers

10. 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

11. unimatix

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

12. dan815

13. 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

14. dan815

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

15. dan815

16. unimatix

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

17. dan815

isnt simpsons just a higher order trap

18. unimatix

I don't really know what that means.

19. dan815

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

20. dan815

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

21. unimatix

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

22. dan815

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

23. dan815
24. dan815

ok i see now

25. dan815

basically for trapezoidal

26. dan815

|dw:1442549451279:dw|

27. dan815

|dw:1442549479549:dw|

28. dan815

we went from line connections to parabollic connections

29. dan815

for trap to simpsons

30. dan815

u can keep going on in this way

31. dan815

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

32. unimatix

Okay that makes sense