Let \(R_n\) denote an \(n\)-dimensional region and \(w\) a weight function defined on \(R_n\). This paper is concerned with the existence and construction of approximations of the type \(\int_{R_n} wf\cong \sum_{k=1}^N A_kf(\mu_k)\), where the approximation is precise for polynomials up to a certain degree. Sufficient conditions are given that common zeros of \(n\) polynomials of degree \(m\), in \(n\) variables, can be used as points of evaluation in a formula having precision \(2m-1\). A subset consisting of more than \(N= \binom{2m-1+n}{n} - n\binom{m-1+n}{n}\) of the common zeros has nonzero weights associated with it. In some instances considerably fewer than \(N\) of the weights are nonzero. Examples of previously known and new formulas are given.