An approach to construct explicit integral representations for two-layer ReLU networks is presented, which provides relatively simple representations for any multivariate polynomial. Quantitative bounds are provided for a particular, sharpened ReLU integral representation, which involves a harmonic extension and a projection. The bounds demonstrate that functions can be approximated with $L^{2}(\mathcal{D})$ errors that do not depend explicitly on dimension or degree, but rather the coefficients of their monomial expansions and the distribution $\mathcal{D}$. We also present a connection to the RKHS of the exponential kernel $K(x,y)=\exp\left(\left\langle x,y\right\rangle \right)$, and a very simple integral representation involving additionally multiplication via a fixed function which has better quantitative bounds.