Let R be the class of all functions that are properly Riemann-integrable on [0, 1], and let IR be the class of all functions that are properly Riemann-integrable on [a, 1] for all a > 0 a > 0 and for which \[ lim a → 0 + ∫ a 1 f ( x ) d x \lim \limits _{a \to {0^+}} \int _a^1 {f(x)\;dx} \] exists and is finite. There are computational schemes that produce a convergent sequence of approximations to the integral of any function in R; the trapezoid rule is one. In this paper, it is shown that there is no computational scheme that uses only evaluations of the integrand, that is similarly effective for IR.