Summary: We study linear estimators for the weighted integral of a stochastic process. The process may only be observed on a finite sampling design. The error is defined in a mean square sense, and the process is assumed to satisfy Sacks-Ylvisaker regularity conditions of order \(r\in \mathbb{N}_0\) [see J. Sacks and D. Ylvisaker, Ann. Math. Statist. 41, 2057-2074 (1970; Zbl 0234.62025)]. We show that sampling at the quantiles of a particular density already yields asymptotically optimal estimators. Hereby we extend the results of Sacks and Ylvisaker for regularity \(r=0\) or 1, and we confirm a conjecture by R. L. Eubank, P. L. Smith and P. W. Smith [Ann. Stat. 10, 1295-1301 (1982)].