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I'd like to look at this problem in terms of the definite integral $I = \int_0^5 e^{\sin\sqrt x}dx$, and in terms of the Midpoint Rule. (Then, hopefully, I'll be able to figure out the left-point rule, right-point, Simpson's, and Trapezoidal.)

When the number of intervals increases by a factor of $q$, the approximation error decreases by a factor of $r(q)$, where $r$ depends on the particular method (let's try the Midpoint Rule). How do you determine the function $r$ theoretically?

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You're aware of Euler-Maclaurin, by any chance? Is this homework, or are you doing this on your own? (I'm trying to gauge what sort of answer to give you.) –  J. M. Oct 28 '11 at 7:37
    
    
I am not familiar with Euler-Maclaurin. I'm in the middle of learning the Taylor series (and the Maclaurin series)... thoughts on giving me some direction? –  dmonopoly Nov 2 '11 at 4:11
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Okay. I figured out that the whole idea is applying the error bounds formula by subbing in "qn" for "n" and seeing how the resulting error changes.

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