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?