Let $\mu$ be a continuous mean regression function defined on $U$, the unit cube in $N$-dimensional Euclidean space. Let $F$ be a distribution function with support in $U$, and let $M$ denote the indefinite integral of $\mu$ with respect to $F$. This paper provides consistency results, including rates of convergence, for a certain estimator of $M$ in the case that the $n$th estimate is based on observations at points $\mathbf{t}_{n1},\cdots, \mathbf{t}_{nn}$ of $U$. The estimator is the $N$-dimensional analogue of that considered by Brunk (1970).