# Midpoint Rule, Trapezoidal Rule, etc.: When the number of intervals increases by a factor of $q$, the approximation error decreases by $r(q) =\;$?

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?

• 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. is a poor mathematician Oct 28 '11 at 7:37
• en.wikipedia.org/wiki/Rectangle_method#Error – Peđa Terzić Oct 28 '11 at 8:02
• 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